Crowdsourcing New Product Design on the Web: An Analysis of

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 248170 13 pageshttpdxdoiorg1011552013248170

Research ArticleCrowdsourcing New Product Design on the WebAn Analysis of Online Designer Platform Service

Xin Dai1 Pui-Sze Chow2 Jin-Hui Zheng2 and Chun-Hung Chiu1

1 Sun Yat-Sen Business School Sun Yat-Sen University 135 West Xingang Road Guangzhou 510275 China2 Institute of Textiles and Clothing The Hong Kong Polytechnic University Kowloon Hong Kong

Correspondence should be addressed to Pui-Sze Chow lpschowoutlookcom

Received 2 August 2013 Revised 7 October 2013 Accepted 8 October 2013

Academic Editor Kannan Govindan

Copyright copy 2013 Xin Dai et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A designer is a core resource in the fashion industry Successful designers need to be creative and quick to understand the businessand wider environment in which they are operating The Designer Platform Service (DPS) which combines the mechanism ofcrowdsourcing and group buying on the web provides a platform for entrant designers to try their abilities in the real marketpractice Freelance designers post design samples or sketches of products on the website of DPS and consumers may preorder theproducts (each at a fixed price) online based on the design information Once the number of ordering reaches or passes a certainthreshold that is the minimum production quantity (MPQ) DPS will arrange for production and delivery according to the ordersreceived This novel service boosts the growth of entrant designers and links designing works with real markets directly We areinterested in how the price and MPQ decisions are made in DPS with consideration of the entrant designerrsquos objective decisionsequences and customer demand structures We develop Stackelberg games to model and derive the equilibrium solutions underindividual scenarios Our findings suggest feasibility of the DPS business model

1 Introduction

Fashion is among the most important creative industries thatenjoy a great deal of attention these days Fashion designersare the core of value creation in the fashion industry asthey are normally the source of creativity In the UK alonethe designer sector produced pound700 million income in 2003of which pound75 million (11 percent) was from license fees todesigners [1]The fashion industry is also a highly competitiveindustry where product life cycles are short economiesgained by product differentiation are built on brand imageand product styling can be quickly imitated [2] Designershave to play a dual role as creative individuals and asentrepreneurs in any provided complicated situation

Entrant designers are an important source of innovationin the fashion industry Yet the professional lives of fashionentrant designers are particularly hard It takes time andresources for advertising and promotion so that a newdesigner label can sustain a share in the market whereasmost new start-up designers are lacking those resources [3]

However in themarket practice resources are often allocatedon those established designers as the market has provedthe attractiveness of their products so that the resourcesallocated could be expected with higher and less risky returnWithout advertising and promotion most entrant designerswould start their career with a relatively low and uncertaindemand for their design until they build up their name andcultivate a market The low and uncertain demand faced byentrant designers hinders their design from being carriedto production as this demand may be far lower than theminimum production quantity which is commonly appliedin textilemanufacturing Even if the production can bemadethe small production quantity and uncertain demand wouldlead to high production and inventory cost

In the past decade the internet has bred a new approachcalled rdquocrowdsourcingrdquo that help individuals make theirinnovative ideas feasible Named byHowe [4] crowdsourcingrefers to the method that solicits contributions from a largegroup of people fromanonline community in order tomake acertain service idea or product available or feasible Howe [4]

2 Mathematical Problems in Engineering

also defined the sponsoring crowd as ldquothe new pool of cheaplabor everyday people using their spare cycles to createcontent solve problems and even do corporate RampDrdquo Manyworks have been done recently on crowdsourcing contests[5] and noncompetitive crowdsourcing ideation initiatives[6ndash8] For example the crowdsourcing t-shirt platformteespringcom provides a way for people to create and sell t-shirts online to raisemoneyHere is how it works first peopledesign a t-shirt and post it on the website choose a goal andset a price to launch a campaign page second people sendthe campaign page to supporters and let them preorder the t-shirts towards the campaign goal (preordering is free buyerswill only be charged if the goal is reached) third peoplecan continue to sell shirts past the goal until the campaignends Once it does the platform handle production andshipment people will get a check for the profit The businessmodel of teespringcom which combines the mechanism ofcrowdsourcing and group buying on the web has provideda platform for entrant designers to try their abilities in thereal market practice In this business model there are twokey decisions to make the price of the product and the targetvolume for production (equal to the campaign goal in the caseof teespringcom) which are similar to the widely discussedproblem of ldquojoint pricing-production decisionsrdquo in operationmanagement [9 10]

To better illustrate how this business model workswe consider a simplified case Designer Platform Service(DPS) which works as a retailer connecting designers withcustomers and designers with manufacturers Similarly DPScrowdsources the product design tasks to the entrant designercommunity and allows the customers to make purchasesin the way of group buying First entrant designers postdesign samples or sketches of products on the website ofDPS and then consumers preorder the product (each at afixed price) online based on the product information Theentrant designer decides the selling price of the productwhile the DPS decides the minimum production quantity(MPQ) Only when the preorder quantity reaches or passesthe MPQ threshold will DPS arrange for production anddelivery Preorder customers then make their payment andget the products delivered Otherwise no production willbe arranged and no one will be charged This novel serviceprovides variety to customers boosts the growth of entrantdesigners and links designing works with real markets Inthe whole process we are most interested in how the priceand MPQ decisions are made within DPS with considera-tion of entrant designerrsquos objective decision sequences andcustomer demand structures

Through the study in the form of Stackelberg games andby considering various scenarios we investigate the decisionsin themechanismwith B2C transactions an entrant designerutilizes DPS to promote her design to consumer and wereveal the price-discovery mechanism in this e-market Weconsider that in our model the two key players the entrantdesigner and DSP have very different objectives Due to thespecial stage of an entrant designer in her professional careerher priority mission is to leverage the creativity with marketacceptance so that she can later find out a way to boost herdesign to profitability Our findings provide insights on how

the mechanism protects DPS in pricing decision in a highlyversatile and risky market and how the mechanism providesa feasible path for entrant designers to test and adjust tothe market with limited loss Moreover our findings suggestfeasibility of the DPS business model

The paper contributes in providing insights of equilib-rium decisions in the combined business model of crowd-sourcing and group buyingThe paper is new in investigatinga creative business model which could significantly utilizecreativity and reduce cost The paper is also innovative inconsidering business players having nonmonetary objectivesBy investigating the two key decisions in the DSP we showhow an online platform could raise a business by crowdsourc-ing designs arranging manufacturing and selling to groupbuying customers Specifically we derive optimal decisionsfor the twoparties under the different objectives of the entrantdesigner decision sequences and market demands throughwhich we show how an entrant designer could benefit fromDSP The findings provide guidelines to practitioners inplatform service and designers and help utilize creation withprofit

The remainder of the paper is organized as follows InSection 2 we review the related literature We construct theStackelberg game models in Section 3 and find the bestresponse decisions in Section 4 In Section 5 we derive theequilibrium decisions of the games with specific demandfunctions We discuss and conclude in Section 6 with limi-tations and future research directions

2 Literature Review

This paper relates to the literature (eg [11ndash13]) in serviceplatform and group buying business on the web Web-basedgroup buying mechanisms are being widely used in bothbusiness-to-business (B2B) and business-to-consumer (B2C)transactions Li and Lee [12] show that with the uncertainpeer-produced services quality a monopolistic platformprovider has no incentive in offering multiple quality classesof service while two competing platform providers may offeridentical service contracts but still receive nonnegative profitAnand and Aron [14] studied a monopolist offering Web-based group buying under different kinds of demand uncer-tainty They derive the monopolistrsquos optimal group-buyingschedule under varying conditions of heterogeneity in thedemand regimes and compare its profits with those obtainedunder the more conventional posted-price mechanism

Pricing decision and determination of theminimum pro-duction quantity with price-dependent demand are the mainfocus of the models explored in this paper In marketing andoperations management literature there is a huge amount ofrelated studies under various settings and we review someof them as follows For instance Pasternack [15] studies thepricing decision and return policy for perishable productsCai et al [16] explore from the game-theoretic perspectivethe pricing scheme of a supply chain with dual channels thatcompete with one another The authors show that a simpleprice discount contract can achieve channel coordination Inparticular the pricing issue has aroused the attention ofmany

Mathematical Problems in Engineering 3

researchers when the demand is price-dependent Petruzziand Dada [17] investigate the optimal selling price andstocking quantity under the newsvendor domain with price-dependent demand Yang et al [18] consider an inventorysystem for noninstantaneous deteriorating items with price-dependent demand and develop an algorithm to solve thecorresponding optimal price and order quantity when partialbacklog is allowed Chen et al [19] suggest some coordinationschemes for a supply chain with lead time consideration andprice-dependent demand Chiu et al [20] suggest a mecha-nism for coordinating the pricing and stocking decisions fora supply chain with price-dependent demand

To make use of the economy of scale it is common forsuppliers to impose some minimum quantity requirementon orders Such practice is prevalent in the apparel industryand Fisher and Raman [21] provide a well-studied exampleof minimum order quantity imposition in the industry Theclassical inventory literature studies the minimum quantityin the form of lot sizing economic order quantity and batchordering problems (see eg [22]) Recently Chow et al[23] investigate how imposition of minimum order quan-tity affects a fashion supply chain adopting quick responsestrategy Their findings suggest that the order constraint hasdifferent impacts on different supply chain agents and thusshould be set carefully for the sake of the whole supply chain

3 Problem Description

We consider the situation that an online retailer (119877) providesa designer platform service (DPS) to an entrant designer (119863)to display one of her (throughout the paper we employ thepronoun ldquosherdquo for the designer and ldquoherdquo for the retailer forease of presentation) own-designed apparel to customersTheretail price of the product 119901 gt 0 is endogenous and decidedby 119863 Customers who are willing to buy the product willpreorder at 119901 online before a specified deadline of orderingThe demand of the product which is reflected by the totalquantity of the preorders placed is price-dependent and isgiven by

119910 (119901 119909) = 119911 (119901) 119909 (1)

where 0 le 119911(119901) lt +infin and 119889119911(119901)119889119901 = 1199111015840(119901) lt 0 for all

119901 ge 0 and 119909 ge 0 is a randomvariable with probability densityfunction 119891(119909) and cumulative distribution function 119865(119909)Without further notification we assume that 119889119865(119909)119889119909 =

119891(119909) gt 0 for all 119909 ge 0The total production cost with production quantity 119902 is

given by

119862 (119902) = 119892 + ℎ119902 (2)

where 119892 gt 0 is the fixed cost and ℎ gt 0 is the variable cost ofproduction We note that the unit product cost is given by

119888 (119902) =119862 (119902)

119902=

119892

119902 + ℎ (3)

and hence119889119888 (119902)

119889119902= minus119892119902

minus2lt 0 forall119902 gt 0 (4)

Specifically 119888(119902) is strictly decreasing in 119902 for all 119902 gt 0To justify the fixed cost of production119877 requires a minimumproduction quantity (MPQ) 119876 gt 0 such that if 119910(119901 119909) ge 119876then119877will proceed to arrange amanufacturer for productionand all the orders placed will be fulfilled and the customerspay for the products Otherwise the product will not beproduced and customers will not be chargedMathematicallythe production quantity 119902 is given by

119902 (119901 119876 119909) = 119910 (119901 119909) if 119910 (119901 119909) ge 1198760 ow

(5)

We consider that the profit of selling the product is sharedby119877 and119863with the former sharing120572 whilst the latter sharing(1 minus 120572) of the total profit where 0 le 120572 le 1 As a remarkdifferent ways of profit allocation under DPS project couldresult in different optimal solutions of119876 and 119901 However theeffects of the allocation schemes are out of the scope of thispaper

For any given 119876 gt 0 and 119901 gt 0 the profit of 119877 is given by

120587 (119876 119901 119909)

=

120572 [119901119910 (119901 119909) minus 119862 (119910 (119901 119909))] if 119910 (119901 119909) ge 119876

0 ow

= 120572 [(119901 minus ℎ) 119911 (119901) 119909 minus 119892] if 119909119911 (119901) ge 1198760 ow

(6)

Taking the expectation of 120587(119876 119901 119909) on 119909 the expectedprofit of 119877 is given by

119864 [120587 (119876 119901)] = 120572int

infin

119876119911(119901)

(119901 minus ℎ) 119911 (119901) 119909 minus 119892119891 (119909) 119889119909 (7)

The feasibility of DPS depends on whether the total orderquantity received is greater than the MPQ We define thefeasibility probability of DPS as the probability of the totalorder quantity being greater than the MPQ that is

Ψ (119901119876) = Pr119909 ge 119876

119911 (119901) = 1 minus 119865[

119876

119911 (119901)] (8)

The sales volume is given by

119902 (119901 119876 119909) = 119910 (119901 119909) = 119909119911 (119901) if 119909119911 (119901) ge 1198760 ow

(9)

Correspondingly the expected sales volume is given by

119864 [119902 (119901 119876)] = 119911 (119901)int

infin

119876119911(119901)

119909119891 (119909) 119889119909 (10)

As a remark 119877 must be profitable in order for himto participate in DPS In other words we need to have120587(119876 119901 119909) gt 0 for all 119909 ge 119876119911(119901) which is equivalent to120572[(119901 minus ℎ)119911(119901)119909 minus 119892] gt 0 for all 119909 ge 119876119911(119901) Simplifying


Table 1 The list of scenarios being considered in this paper

Scenario Decision sequence Followerrsquos problem Leaderrsquos problemLeader Follower

Project-orientedR-led Retailer Project-oriented

designer

For any given Q to find119901lowast= argmax119901ge0

Ψ(119901 119876)

st 120572120587(119876 119901) ge 0 for all 119909 ge 0

Anticipating 119901lowast to find119876lowast= argmax119876ge0

119864[120587(119876 119901lowast)]

Market-orientedR-led Retailer Market-oriented

designer


119864[119902(119901 119876)]

st 120572120587(119876 119901) ge 0 for all 119909 ge 0


119864[120587(119876 119901lowast)]

Project-orientedD-led

Project-orienteddesigner Retailer

For any given 119901 to find119876lowast= argmax119876ge0

119864[120587(119876 119901)]


Ψ(119901 119876lowast)

st 120572120587(119876lowast 119901lowast) ge 0 for all 119909 ge 0

Market-orientedD-led

Market-orienteddesigner Retailer


119864[120587(119876 119901)]


119864[119902(119901 119876lowast)]


the above yields (119901 minus ℎ)119911(119901)119909 ge 119892 for all 119909 ge 119876119911(119901)By putting 119876 = 119909119911(119901) we need to have (119901 minus ℎ)119876 ge 119892 or119901 ge ℎ + 119892119876 Moreover for any given119876 and 119901 if 119909 lt 119876119911(119901)for all 119909 ge 0 then both 119863 and 119877 have zero profit Thereforethere should exist some 119909 such that 119909 ge 119876119911(119901)

For any given pair of (119876 119901) the pair of (119876 119901) is said to befeasible if (i) 119901 ge ℎ + 119892119876 and if (ii) there exists some 119909 suchthat 119909 ge 119876119911(119901) The following summarize condition (i) of(119876 119901) being feasible Let

1199010(119876) = ℎ +

119892

119876(11)

(C1) 119901 ge 1199010(119876) gt ℎ for any given 119876

For condition (ii) the specific condition depends on theexact form of 119891(119909)

4 The Best Response Decisions of DPS

There are two decision variables under DPS namely119876 and 119901The optimal values of119876 and 119901may be different with differentdecision sequences Therefore in this paper we consider twoscenarios one with 119877 making the decision first whilst theother with 119863 being the first mover A game with 119877 as thefirst mover is referred to as an 119877-led (game) whereas the onewith 119863 as the first mover is referred to as a 119863-led (game)The objectives of the two parties are also different 119877is tomaximize his expected profit which must be nonnegativeFor119863 whether her product can be sold or not or how manycustomers will buy her product are more important than theprofit she is to earn from selling her product Therefore weconsider the following two types of119863whose objectives are (1)maximizing the feasibility probability of DPS project and (2)maximizing the expected market size which is equivalent tomaximizing the expected sales volumeWe refer to119863 havingthe first objective as ldquoProject-oriented Designerrdquo whilst theone with the second objective is ldquoMarket-oriented DesignerrdquoTable 1 summarizes the scenarios that are considered in thispaper

As 119863 and 119877 make decisions sequentially we apply theStackelberg game in analyzing the model in which the firstmover acts as the game leader and the late mover acts asthe game follower Moreover we apply backward inductionto obtain the optimal solutions of the players in the gamePropositions 1 to 4 provide the best response solutions tovarious scenarios

Proposition 1 Under the ldquoProject-oriented 119877-ledrdquo scenario(a) for any fixed 119876 gt 0 the DPS feasibility probability

Ψ(119876 119901) is decreasing in 119901 for 119901 gt 0 and the bestresponse price that maximizes Ψ(119876 119901) is uniquelygiven by 119901lowast

1(119876) = 119901

0(119876)

(b) if the best response MPQ that maximizes the expectedprofit of R (denoted by119876

1

lowast) exists and satisfies the first-order optimality condition it is given by

1198761

lowast= arg119876 ge 0 119876119911(ℎ +

119892

119876) + 119892119911

1015840(ℎ +

119892

119876) = 0

(12)

Proof All proofs are relegated in the Appendix

Proposition 2 Under the ldquoMarket-oriented 119877-ledrdquo scenario(a) for any fixed 119876 gt 0 the expected sales volume

119864[119902(119876 119901)] is decreasing in 119901 for 119901 gt 0 and the bestresponse price that maximizes the expected market sizeis uniquely given by

119901lowast

2(119876) = 119901

0(119876) (13)

(b) if the best response MPQ that maximizes the expectedprofit of119877 (denoted by119876

2


1198762

lowast= arg119876 gt 0 119876119911(ℎ +

119892

119876) + 119892119911

1015840(ℎ +

119892

119876) = 0

(14)


Propositions 1 and 2 assert that when 119877 is the first moverin DPS the best response decisions are the same whether119863 isproject-oriented or market-oriented In particular under thetwo 119877-led scenarios it is always optimal for119863 to set the retailprice as low as possible [ie 119901

0(119876)] Such result is natural as

the objective of 119863 is not related to the expected profit she isto earn from DPS

Proposition 3 Under the ldquoProject-oriented119863-ledrdquo scenario

(a) for any fixed 119901 gt 1199010

(a-i) 119864[120587(119876 119901)] is strictly concave in 119876 if and only if

(119901 minus ℎ) 119911 (119901) 119891 [119876

119911 (119901)] + [(119901 minus ℎ)119876 minus 119892]119891

1015840[

119876

119911 (119901)] gt 0

forall119876 gt 0

(15)

(a-ii) a sufficient condition for strict concavity of119864[120587(119876 119901)] is 1198911015840[119876119911(119901)] ge 0 for all 119876 gt 0 and

(a-iii) if 1198911015840[119876119911(119901)] ge 0 the best response MPQ thatmaximizes the expected profit of 119877 is uniquelygiven by 119876

3

lowast(119901) = 119892(119901 minus ℎ)

(b) if the best response price that maximizes the feasibilityprobability of DPS exists and satisfies the first-orderoptimality condition it is given by

1199013

lowast= arg 119901 gt 119901

0 119911 (119901) + (119901 minus ℎ) 119911

1015840(119901) = 0 (16)

Proposition 4 Under the ldquoMarket-oriented119863-ledrdquo scenario



(119901 minus ℎ) 119911 (119901) 119891 [119876

119911 (119901)] + [(119901 minus ℎ)119876 minus 119892]119891

1015840[

119876

119911 (119901)] gt 0

forall119876 gt 0

(17)

(a-ii) a sufficient condition for strict concavity of119864[120587(119876 119901)]is 1198911015840[119876119911(119901)] ge 0 for all 119876 gt 0 and

(a-iii) if 1198911015840[119876119911(119901)] ge 0 the best response MPQ thatmaximizes the expected profit of 119877 is uniquelygiven by

1198764

lowast(119901) =

119892

119901 minus ℎ(18)

(b) if the best response price that maximizes the expectedsales volume exists and satisfies the first-order optimal-ity condition it is given by

1199014

lowast= arg 119901 gt 119901

0 120577 (119901) = 0 (19)

where

120577 (119901) =(119901 minus ℎ) 1199111015840(119901) ((119901 minus ℎ)

2

1199112(119901)int

infin

119892[(119901minusℎ)119911(119901)]

119909119891 (119909) 119889119909

+ 1198922119891[

119892

(119901 minus ℎ) 119911 (119901)])

+ 1198922119911 (119901) 119891(

119892

(119901 minus ℎ) 119911 (119901))

(20)

On the other hand Propositions 3 and 4 indicate thatwhen 119863 is the first mover in DPS (ie the two 119863-ledscenarios) if the best response MPQ exists and is positivethen we have 119876

3

lowast(119901) = 119876

4

lowast(119901) = 119892(119901 minus ℎ)

As a remark it is the lowest MPQ that promises nonneg-ative profit In addition the smaller the MPQ the higher theunit price that119863 shouldmeet whichmeans more profit fromexcess order toMPQTherefore regardless of the objective of119863 the best response function of 119877 is given by

1198760(119901) =

119892

119901 minus ℎ (21)

Notice that119892 is the fixed cost for production whilst (119901minusℎ)can be viewed as the unit ldquoprofit marginrdquo for each piece ofapparel under DPS In other words the retailer should set theMPQ in the way that such a corresponding profit can justifythe fixed production cost

Moreover we note that the existence of the best responseretail price and the best response MPQ depend on thefunctions 119911(119901) and 119891(119909) there may be situations where nooptimal solutions exist There is no general condition toensure the existence of them Propositions 1 to 4 show theanalytical forms of the optimal retail price and the optimalMPQ in case they exist

5 Analysis with Specific Demand Functions

Toobtainmore insight we consider specific forms of 119911(119901) and119865(119909) in this section Specifically we consider that 119909 follows auniform distribution 119880[119897 119906] where 0 lt 119897 lt 119906 For 119911(119901) weconsider two specific types that are commonly adopted in theliterature (eg [17]) namely

(1) additive form

119911119860(119901) =

119886 minus 119887119901 for 0 lt 119901 lt 119886119887

0 for 119901 ge 119886119887

(22)

for 119886 gt 119887ℎ gt 0 (as 119901 gt ℎ if 119886 gt 119887ℎ does not hold then119901 gt 119886119887 and 119911

119860(119901) = 0) and

(2) multiplicative form 119911119872(119901) = 119886119901

minus119887 for 119886 gt 0 and 119887 gt0


Table 2 Equilibrium decisions with additive form demand function 119911119860(119901) = 119886 minus 119887119901

Scenario Condition(s) for existence of the equilibrium Equilibrium 119876 Equilibrium 119901

Project-oriented R-ledMarket-oriented R-led 119892 lt (119886 minus 119887ℎ)

2

1199064119887 2119887119892(119886 minus 119887ℎ) (119886 + 119887ℎ)2119887

Project-oriented D-ledMarket-oriented D-led 119892 lt (119886 minus 119887ℎ)

2

1199064119887 and 1199011198714119860

lt 1199014119860lt 1199011198674119860

119892(1199014119860minus ℎ) Given by 119901

4119860

Next we study the equilibriums with two demand formsseparately

51 Additive Form Demand Function With 119911(119901) = 119911119860(119901) =

119886minus119887119901 Propositions 5 to 7 detail the equilibriumdecisions andthe conditions for the existence of the equilibrium decisionsfor DPS under different scenarios

Proposition 5 For the two ldquo119877-ledrdquo scenarios

(a) there exists feasible (119876 119901) only if 119892 lt (119886 minus 119887ℎ)21199064119887

and 119876 gt 119892119887(119886 minus 119887ℎ)(b) for 119892 lt (119886 minus 119887ℎ)

21199064119887 the expected profit of 119877 is

unimodal in 119876 for all 119876 gt 0 and the equilibriumMPQ is uniquely given by 119876

1119860

lowast= 2119892119887(119886 minus 119887ℎ) gt

119892119887(119886 minus 119887ℎ) Correspondingly the equilibrium price isuniquely given by 119901

1119860

lowast= (119886 + 119887ℎ)2119887 gt ℎ

LetΔ = 119906[119906(119886minus119887ℎ)2minus4119887119892]1199011198713119860

= (119886+119887ℎ)2119887minusradicΔ2119906119887and 119901

1198673119860= (119886 + 119887ℎ)2119887 + radicΔ2119906119887

Proposition 6 For the ldquoProject-oriented119863-ledrdquo scenario

(a) the best response MPQ of 119877 is given by1198760(119901) = 119892(119901minus

ℎ) and the pair (1198760(119901) 119901) is feasible only if 119892 lt 119906(119886 minus

119887ℎ)24119887

(b) for 119892 lt 119906(119886 minus 119887ℎ)24119887 the DPS feasibility probabilityis strictly concave in 119901 the equilibrium retail price is1199013119860

lowast= (119886 + 119887ℎ)2119887 gt ℎ and the equilibrium MPS is

1198763119860

lowast= 2119887119892(119886 minus 119887ℎ)

Proposition 7 For the ldquoMarket-oriented 119863-ledrdquo scenario for119892 lt 119906(119886 minus 119887ℎ)

24119887

(a) the expected sales volume function 119864[119902(119901 1198760(119901))]

given by (10) is strictly concave in 119901(b) let119901

4119860= arg1199011198922(2119886+119887ℎminus3119887119901)minus119887119906

2(119901minusℎ)

3(119886minus119887119901)

2=

0 If 1199011198713119860

lt 1199014119860lt 1199011198673119860

then the equilibrium 119901 is1199014119860

and the equilibrium MPS is 1198764119860

lowast= 119892(119901

4119860minus ℎ)

otherwise (1198760(119901) 119901) is not feasible for all 119901 gt 0

Table 2 summarizes the equilibrium decisions undervarious scenarios and the respective conditions required

With 119911(119901) = 119911119860(119901) = 119886 minus 119887119901 we found that 119892 minus (119886 minus

119887ℎ)21199064119887 lt 0 is the common condition for the existence

of the equilibrium solutions for all scenarios As 119892 minus (119886 minus119887ℎ)21199064119887 lt 0 is strictly increasing with 119892 ℎ and 119887 and

is strictly decreasing with 119886 and 119906 the condition 119892 minus (119886 minus119887ℎ)21199064119887 lt 0 implies that the equilibrium decision may

not exist when 119892 ℎ andor 119887 are big andor 119886 andor 119906 aresmall We argue such results as follows When 119892 andor ℎare big it is not profitable for 119877 under DPS and hence 119877will not participate with DPS When 119887 is big the productdemand is very sensitive to the price such that it is notpossible to set a price to make 119902 no less than MPQ and 119877with profitable When 119886 andor 119906 are small the demand baseis small Therefore again it is not possible to set a price tomake 119902 no less than MPQ and 119877 profitable

For the equilibrium decisions they are the same for thescenarios Project-oriented119877-ledMarket-oriented119877-led andProject-oriented 119863-led However the equilibrium decisionsof the Market-oriented 119863-led scenario are different from theother three scenarios Moreover for the Market-oriented 119863-led scenario there is an extra condition 119901

1198714119860lt 1199014119860

lt

1199011198674119860

for the existence of the equilibrium solutions Thiscondition gives an upper bound and a lower bound of theequilibrium 119901 We argue such results as follows Being anentrant designer 119863 cares less about maximizing her ownprofit Rather she aims at either maximizing the expectedsales volume or the feasibility probability of the DPS Bothobjectives can be fulfilled by maximizing the demand whichis also favorable to the profit-maximizing 119877 In other wordsthe objective of 119863 is consistent with that of 119877 thereforethe effect of decision sequence becomes negligible Besideswhether 119863 is project- or market-oriented it is also optimalfor her to set 119901 as small as possible to maximize the marketsize whilst keeping 119877 profitable From the perspective of 119877as discussed in the previous section the optimal MPQ is theone that barely justifies the fixed production costThis reflectsthat it is always optimal for him to make DPS feasible sothat he can have the opportunity to gain profit Hence theequilibrium decisions are almost the same under differentdecision sequences and different objectives of 119863 when thereis no limitation on setting 119901 However because of the extracondition for the equilibrium 119901 the equilibrium decisionsfor theMarket-oriented119863-led scenario are different from theother three scenarios

52 Multiplicative Form Demand Function With 119911(119901) =

119911119872(119901) = 119886119901

minus119887 Propositions 8 to 10 detail the equilibriumdecisions and the conditions for the existence of the equilib-rium decisions for DPS under different scenarios


(a) the best response retail price of 119863 is given by 1199010(119876) =

ℎ + 119892119876(b) for 0 lt 119887 le 1 the expected profit of 119877 119864[120587

1119872(119876)] is

strictly decreasing in 119876 and 119864[1205871119872(0)] = +infin and


Table 3 Equilibrium decisions with multiplicative form demand function 119911119872(119901) = 119886119901

minus119887

Scenario Condition(s) for existence of equilibrium Equilibrium 119876 Equilibrium 119901

Project-oriented R-led119887 gt 1 and 119892(119887 minus 1)[ℎ119887(119887 minus 1)]119887119886ℎ lt 119906

(119887 minus 1)119892ℎ 119887ℎ(119887 minus 1)Market-oriented R-ledProject-oriented D-led

119887 gt 1 and 119897 lt 119892(119887 minus 1)[ℎ119887(119887 minus 1)]119887119886ℎ lt 119906Market-oriented D-led

(c) for 119887 gt 1 the expected profit of the retailer is unimodalin 119876 for all 119876 gt 0

(c-i) if 119892(119887 minus 1)[ℎ119887(119887 minus 1)]119887119886ℎ ge 119906 (119876 1199010(119876)) is

infeasible for all 119876 gt 0 and(c-ii) otherwise the equilibriumMPQ and the equilib-

rium 119901 are given by 1198761119872

lowast= (119887 minus 1)119892ℎ and

1199011119872

lowast= 119887ℎ(119887 minus 1) respectively

Let Δ3119860

= 119906[119906(119886 minus 119887ℎ)2minus 4119887119892] 119901

1198713119860= (119886 + 119887ℎ)2119887 minus

radicΔ31198602119906119887 and 119901

1198673119860= (119886 + 119887ℎ)2119887 +radicΔ

31198602119906119887 Moreover

let

1198703119872(119901) =

119892119901119887

119886 (119901 minus ℎ)lt 119906 (23)

and denote 1199013119872119897

1015840 1199013119872119906

1015840 1199013119872119906

10158401015840 1199013119872119906

101584010158401015840 1199013119872119897

10158401015840 1199013119872119897

1015840101584010158401199013119872119897

1015840 and 1199013119872119906

1015840 that satisfy (i) 1199013119872119897

1015840gt 1199013119872119906

1015840gt ℎ (ii)

1198703119872(1199013119872119897

1015840) = 119897 and (iii) 119870

3119872(1199013119872119906

1015840) = 119906 and 119901

3119872119906

101584010158401199013119872119906

101584010158401015840 1199013119872119897

10158401015840 and 1199013119872119897

101584010158401015840 satisfy (i) ℎ lt 1199013119872119906

10158401015840lt 1199013119872119897

10158401015840le

1199013119872119897

101584010158401015840lt 1199013119872119906

101584010158401015840 (ii) 1198703119872(1199013119872119906

10158401015840) = 119870

3119872(1199013119872119906

101584010158401015840) = 119906 (iii)

1198703119872(1199013119872119897

10158401015840) = 119870

3119872(1199013119872119897

101584010158401015840) = 119897 (iv) 119897 lt 119870

3119872(119901) lt 119906 for

all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 or 1199013119872119897

101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 and (v)1198703119872(119901) le 119897 for all 119901

3119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840


(a) the best response MPQ of 119877 is 1198760(119901) = 119892(119901 minus ℎ)

(b) for 119887 le 1 the feasibility probability Ψ(119901) is strictlyincreasing in 119901 for all 119901

3119872119906

1015840lt 119901 lt 119901

3119872119897

1015840 and Ψ(119897)=1 for all 119901 ge 119901

3119872119897

1015840(c) for 119887 gt 1

(c-i) if (119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))119887 ge 119906 then the pair(119876lowast3119872(119901) 119901) is infeasible

(c-ii) if 119897 lt (119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))119887 lt 119906 then Ψ(119901)is unimodal in119901 and ismaximized at119901 = ℎ119887(119887minus1)

(c-iii) if (119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))119887 lt 119897 then Ψ(119901) isstrictly increasing in 119901 for all 119901

3119872119906

10158401015840lt 119901 lt

1199013119872119897

10158401015840 is strictly decreasing in p for all 1199013119872119897

101584010158401015840lt

119901 lt 1199013119872119906

101584010158401015840 andΨ3119872(119901) = 1 for all 119901

3119872119897

10158401015840lt 119901 lt

1199013119872119897

101584010158401015840

Proposition 9 asserts that for the Project-oriented 119863-ledscenario the equilibrium decision exists only if 119887 gt 1 and119897 lt (119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))

119887lt 119906 and the equilibrium 119876

and equilibrium 119901 are given by (119887 minus 1)119892ℎ and 119887ℎ(119887 minus 1)respectively


(a) for 119887 le 1 the expected sales volume 119864[119902(119901 1198760(119901))]

is strictly increasing in 119901 for all 1199013119872119906

1015840lt119901lt119901

3119872119897

1015840 and119864[119902(119901 119876

0(119901))] = (119906 minus 119897)2 for all 119901 ge 119901

3119872119897

1015840(b) for 119887 gt 1

(b-i) if 119897 lt (119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))119887

lt 119906then 119864[119902(119901 119876

0(119901))] is unimodal in 119901 and is

maximized at 119901 = ℎ119887(119887 minus 1) and(b-ii) if (119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))

119887lt 119897 then

119864[119902(119901 1198760(119901))] is strictly increasing in p for all

1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 is strictly decreasing in pfor all 119901

3119872119897

101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 andΨ3119872(119901) = 1 for

all 1199013119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840

Proposition 10 asserts that for the Market-oriented119863-ledscenario the equilibrium decision exists only if 119887 gt 1 and119897 lt (119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))



Table 3 summarizes the best response solutions undervarious scenarios and the respective conditions required

With 119911(119901) = 119911119872(119901) = 119886119901

minus119887 we found that the optimalsolutions are almost the same whether 119863 is project- ormarket-oriented They are also independent of the decisionsequence Similar to the case with 119911(119901) = 119911

119860(119901) = 119886 minus 119887119901

we argue such results as follows Being an entrant designer119863 cares less about maximizing her own profit Rather sheaims at either maximizing the expected sales volume or thefeasibility probability of the DPS Both objectives can befulfilled by maximizing the demand which is also favorableto the profit-maximizing 119877 In other words the objective of119863 is consistent with that of 119877 therefore the effect of decisionsequence becomes negligible Besides whether 119863 is project-or market-oriented it is also optimal for her to set 119901 as smallas possible to maximize the market size whilst keeping 119877profitable Hence the optimal solutions are almost the sameunder different decision sequences and different objectives of119863 From the perspective of 119877 as discussed in the previoussection the optimal MPQ is the one that barely justifies thefixed production costThis reflects that it is always optimal forhim tomake DPS feasible so that he can have the opportunityto gain profit

The optimality conditions provide some other insights tothe equilibrium decisions To be specific 119887 gt 1 is requiredin order to have equilibrium decisions for all four scenarios119892(119887 minus 1)[ℎ119887(119887 minus 1)]

119887119886ℎ lt 119906 is required for the two 119877-led

scenarios and 119897 lt 119892(119887 minus 1)[ℎ119887(119887 minus 1)]119887119886ℎ lt 119906 is required for


the two119863-led scenarios As 119897 does not include the equilibrium119876 and the equilibrium119901 the value of 119897 is irrelevant in derivingthe equilibrium decisions for the two 119877-led scenarios but 119897becomes relevant in deriving the equilibrium decisions forthe two 119863-led scenarios Moreover it is more probable anequilibriumdecision could be derived for two119877-led scenariosthan for the two119863-led scenarios

6 Discussion

This paper concerns a service platform provided by an onlineretailer to an entrant designer By formulating the problem asvarious Stackelberg games we first explore the best responseretail price of the entrant designer and the best responseMPQ of the retailer under different designerrsquos objectives anddecision sequences in a general demand function setting andthenwe explore the equilibriumdecisions of the games underdifferent designerrsquos objectives and decision sequences byconsidering two specific demand function setting structuresEssentially the retailer should set the MPQ that justifies thefixed production cost whilst the entrant designer shouldset the retail price as low as possible to enlarge the marketdemand whilst the retail is still profitable

The current paper bares several limitations including butnot limited to the below First this paper considers the simplecase that the product is sold at a fixed price at all quantitieswhereas there can be various ways in organizing the groupselling a possible extension is to explore howdifferent pricingschemes may bring benefits to all parties in the system Forinstance a pricing scheme with quantity discount that isdifferent levels of discounts at different total order quantitiesis a good candidate for future research Second this paperconsiders the systemwith single designer and single platformThe literature in crowdsourcing suggests that competitionamong source providers which is commonly observed inreality is good for resource allocation Accordingly this studycan be extended to explore the effect of competition amongstmultiple designers andor platforms For other extensionsit is interesting to explore the optimal solutions of the DPSwhen a well-established designer who cares about profitmaximization is involved

Appendix

Mathematical Proofs

Note for brevity we omit one of the variables 119901 or 119902 in119864[119902(119901 119876)] 119864[120587(119876 119901)] and Ψ(119901119876) in the proofs

Proof of Proposition 1 Under the ldquoProject-oriented 119877-ledrdquoscenario

(a) for any given 119876 gt 0 the DPS feasibility probability isgiven by Ψ(119901) = 1 minus 119865[119876119911(119901)] and we have

119889Ψ (119901)

119889119901= 119891[

119876

119911 (119901)] [

1198761199111015840(119901)

1199112(119901)

] le 0 as 1199111015840 (119901) lt 0

119891 (119909) ge 0 forall119909 ge 0

(A1)

Thus the optimal retail price that maximizes theexpected sales volume is the smallest possible valueof the retail price which is equal to 119901

0(119876) by (C1)

(b) anticipating that 119863 would set the retail price as1199012

lowast(119876) = 119901

0(119876) the retailerrsquos expected profit when

having 119876 as the MPQ and the corresponding firstderivative in 119876 are given by the below respectively

119864 [120587 (119876)] =120572int

infin

119876119911[1199012

lowast(119876)]

[(119892

119876)119911 (119901

2

lowast

(119876)) 119909 minus 119892]119891 (119909) 119889119909

= 120572119892[119911 (ℎ + 119892119876)

119876int

infin

119876119911(ℎ+119892119876)

119909119891 (119909) 119889119909

+ 119865(119876

119911 (ℎ + 119892119876)) minus 1]

(A2)

119889119864[120587 (119876)]

119889119876= minus(

120572119892

1198763)[119876119911(ℎ +

119892

119876) + 119892119911

1015840(ℎ +

119892

119876)]

times int

infin

119876119911(ℎ+119892119876)

119909119891 (119909) 119889119909

(A3)

By the participation constraint of119877 that is 120587(119876 119901 119909) gt 0 wehave 119864[120587(119876)] gt 0Therefore intinfin

119876119911(ℎ+119892119876)119909119891(119909)119889119909 gt (119876119911(ℎ+

119892119876))(1 minus 119865(119876119911(ℎ + 119892119876))) gt 0 From (A3) we then have119889119864[120587(119876)]119889119876|

119876=1198762

lowast = 0 hArr 1198762

lowast119911(ℎ + 119892119876

2

lowast) + 119892119911

1015840(ℎ +

1198921198762

lowast) = 0

Proof of Proposition 2 Under the ldquoMarket-oriented 119877-ledrdquoscenario

(a) for any given 119876 gt 0 the expected sales volumeis given by 119864[119902(119901)] = 119911(119901) int

infin

119876119911(119901)119909119891(119909)119889119909 and

we have 119889119864[119902(119901)]119889119901 = 1199111015840(119901)[intinfin

119876119911(119901)119909119891(119909)119889119909 +

1198762119891[119876119911(119901)]119911

2(119901)] le 0 as 1199111015840(119901) lt 0 and 119891(119909) ge

0 for all 119909 ge 0 Thus the optimal retail price thatmaximizes the expected sales volume is the smallestpossible value of the retail price which is equal to1199010(119876) by (C1)

(b) same as Proposition 1(b)

Proof of Proposition 3 (a) For any fixed 119901 gt 1199010 the expected

profit of 119877 is given by

119864 [120587 (119876)] = 120572int

infin

119876119911(119901)

[(119901 minus ℎ) 119911 (119901) 119909 minus 119892] 119891 (119909) 119889119909

= 120572((119901 minus ℎ) 119911 (119901)int

infin

119876119911(119901)

119909119891 (119909) 119889119909

+119892119865[119876

119911 (119901)] + 119892)

(A4)


The first and second derivatives of 119864[120587(119876)] wrt 119876 can beeasily derived as follows respectively

119889119864 [120587 (119876)]

119889119876= minus

120572119891 [119876119911 (119901)] [(119901 minus ℎ)119876 minus 119892]

119911 (119901) (A5)

1198892119864 [120587 (119876)]

1198891198762

= minus120572((119901 minus ℎ) 119911 (119901) 119891[119876

119911 (119901)]

+ [(119901 minus ℎ)119876 minus 119892]1198911015840[

119876

119911 (119901)]) times (119911

2(119901))

minus1

(A6)

(a-i) A direct observation from (A6) (a-ii) If1198911015840[119876119911(119901)] ge 0for all 119876 gt 0 then

(119901 minus ℎ) 119911 (119901) 119891 [119876

119911 (119901)] + [(119901 minus ℎ)119876 minus 119892]119891

1015840[

119876

119911 (119901)]

ge (119901 minus ℎ) 119911 (119901) 119891 [119876

119911 (119901)] gt 0

(A7)

Noticed that 119901 ge 1199010(119876) gt ℎ and 119891(119909) gt 0 So by (a-

i) 119864[120587(119876)] is strictly concave in 119876 (a-iii) A direct result bysolving the first-order condition 119889119864[120587(119876)]119889119876 = 0 from(A5)

(b) Anticipating that the retailer sets the minimum pro-cessing quantity as 119876

3

lowast(119901) the probability of DPS feasibility

is given by

Ψ (119901) = 1 minus 119865[1198763

lowast(119901)

119911 (119901)] = 1 minus 119865[

119892

(119901 minus ℎ) 119911 (119901)] (A8)

The first derivative of Ψ(119901) is

119889Ψ (119901)

119889119901= [

119892

(119901 minus ℎ)2

1199112(119901)

] [119911 (119901) + (119901 minus ℎ) 1199111015840(119901)]

times 119891[119892

(119901 minus ℎ) 119911 (119901)]

(A9)

The first-order condition 119889Ψ(1199013

lowast)119889119901 = 0 is equivalent to

119911(1199013

lowast) + (119901

3

lowastminus ℎ)119911

1015840(1199013

lowast) = 0 by direct observation from

(A9)

Proof of Proposition 4 (a) Same as the proofs for Proposition3(a)

(b) Anticipating that the retailer sets the minimumprocessing quantity as 119876

4

lowast(119901) the expected sales volume is

given by 119864[119902(119901)] = 119911(119901) int infin119892[(119901minusℎ)119911(119901)]

119909119891(119909)119889119909It can be easily shown that the first derivative of 119864[119902(119901)]

is in the form

119889119864 [119902 (119901)]

119889119901=

120577 (119901)

[(119901 minus ℎ)3

1199112(119901)]

(A10)

The optimal price that maximizes the expected sales volume(denoted by 119901

4

lowast) if exists should satisfy the first-ordercondition 119889119864[119902(119901

4

lowast)]119889119901 = 0 which is equivalent to 120577(119901) =

0 by direct observation from (A10)

Proof of Proposition 5 (a) From (C1) we need to have119901 ge ℎ+119892119876 By the property of 119911

119860(119901) we also require 0 lt 119901 lt 119886119887

Considering both together we have 119876 gt 119892119887(119886 minus 119887ℎ) Nextthere exists some 119909 such that 119909 ge 119870

1119860(119876) where 119870

1119860(119876) =

119876119911119860(1199011119860

2) Since 119897 lt 119909 lt 119906 we need to have 119870

1119860(119876) lt 119906

which is equivalent to 1198762 minus (119886 minus 119887ℎ)119906119876 + 119887119892119906 lt 0 We proveby contradiction that 119892 le (119886 minus 119887ℎ)21199064119887 Suppose 119892 gt (119886 minus119887ℎ)21199064119887 Then

1198762minus (119886 minus 119887ℎ) 119906119876 + 119887119892119906

gt 1198762minus (119886 minus 119887ℎ) 119906119876 +

(119886 minus 119887ℎ)21199062

4

= [119876 minus(119886 minus 119887ℎ) 119906

2]

2

ge 0

(A11)

A contradiction So we need to have 119892 le (119886 minus 119887ℎ)21199064119887(b) The profit of the retailer when setting MPQ as 119876 is

given by

1205871119860(119876)

=

[1199011119860

lowast(119876) minus ℎ] 119909119911

119860[1199011119860

lowast(119876)] minus 119892 for 119909 ge 119870

1119860(119876)

0 ow(A12)

where 1199011119860

lowast(119876) = 119901

0(119876) = ℎ + 119892119876 by Propositions 1 and 2

Taking expectation we have

119864 [1205871119860(119876)]

= int

119906

1198701119860(119876)

([1199011119860

lowast

(119876) minus ℎ] 119909119911119860[1199011119860

lowast

(119876)] minus 119892) 119891 (119909) 119889119909

=119892 [119906 minus 119870

1119860(119876)]

(119906 minus 119897)([119886 minus 119887ℎ

119876minus119887119892

1198762] [

119906 + 1198701119860(119876)

2] minus 1)

(A13)

The first and second derivative of 119864[1205871119860(119876)] can be easily

derived as follows respectively

119889119864 [1205871119860(119876)]

119889119876=

119892 [1199062minus 1198701119860

2(119876)] [minus (119886 minus 119887ℎ)119876 + 2119887119892]

2 (119906 minus 119897) 1198763

(A14)

1198892119864 [1205871119860(119876)]

1198891198762

=

119892 [1199062minus 1198701119860

2(119876)] [(119886 minus 119887ℎ)119876 minus 3119887119892]

(119906 minus 119897) 1198764

(A15)

From (A15) for 119876 gt 0 1198892119864[1205871119860(119876)]119889119876

2lt 0 hArr 119876 lt

3119887119892(119886 minus 119887ℎ) (as 1199062 gt 1198701119860

2(119876)) Thus 119864[120587

1119860(119876)] is strictly

concave for 119876 in 0 lt 119876 lt 3119887119892(119886 minus 119887ℎ) and is convex for


119876 ge 3119887119892(119886 minus 119887ℎ) From (A14) 119889119864[1205871119860(119876)]119889119876 = 0 hArr

3119887119892(119886 minus 119887ℎ) gt 119876 = 2119887119892(119886 minus 119887ℎ) gt 0 (ie a unique119876 gt 0 satisfying 119889119864[120587

1119860(119876)]119889119876 = 0) Therefore 119864[120587

1119860(119876)]

is unimodal in 119876 for 119876 gt 0 Thus 1198761119860

lowast= 2119887119892(119886 minus 119887ℎ) is

unique Then by direct manipulation 1199011119860

lowast= 1199011119860

lowast(1198761119860

lowast) =

(119886 + 119887ℎ)2119887

Proof of Proposition 6 Under the ldquoProject-oriented 119863-ledrdquoscenario with 119911(119901) = 119911

119860(119901) = 119886 minus 119887119901

(a) by Proposition 3 the optimal MPQ is1198760(119901) = 119892(119901minus

ℎ) which satisfies (C1) Since 119897 le 119909 le 119906 andto ensure (119876

0(119901) 119901) being feasible we need to have

1198760(119901)119911119860(119901) lt 119906 which is equivalent to

120578 (119901) = 1199061198871199012minus 119906 (119886 + 119887ℎ) 119901 + 119906ℎ119886 + 119892 lt 0 (A16)

(b) Suppose 119892 lt 119906(119886minus119887ℎ)24119887 For any given 1199011198713119860

lt 119901 lt

1199011198673119860

the DPS feasibility probability is given by

Ψ3(119901) = 1 minus 119865[

1198763119860

lowast(119901)

119911119860(119901)

]

= minus119892

(119906 minus 119897) (119901 minus ℎ) (119886 minus 119887119901)+ [1 +

119906

(119906 minus 119897)]

(A17)

The first and second derivatives of Ψ3(119901) wrt 119901 can be

derived easily as follows

119889Ψ (119901)

119889119901=

119892

(119906 minus 119897) (119901 minus ℎ)2

(119886 minus 119887119901)2(119886 + ℎ119887 minus 2119887119901) (A18)

1198892Ψ (119901)

1198891199012

= minus2119892

(119906 minus 119897) (119901 minus ℎ)3

(119886 minus 119887119901)3

times [119887 (119901 minus ℎ) (119886 minus 119887119901) + (119886 + ℎ119887 minus 2119887119901)2

] lt 0

(A19)

for all 1199011198713119860

lt 119901 lt 1199011198673119860

So Ψ3(119901) is strictly concave in 119901

for 1199011198713119860

lt 119901 lt 1199011198673119860

Solving the first-order condition from(A18) we have the optimal retail price 119901

3119860

lowast= (119886 + ℎ119887)2119887

which satisfies the constraint as 1199011198713119860

lt 1199013119860

lowastlt 1199011198673119860

Correspondingly the optimal MPS in this case is 119876

3119860

lowast=

1198760(1199013119860

lowast) = 2119887119892(119886 minus 119887ℎ)

Proof of Proposition 7 Under the ldquoMarket-oriented 119863-ledrdquoscenario with 119911(119901) = 119911

119860(119901) = 119886 minus 119887119901

(a) For 119892 lt 119906(119886 minus 119887ℎ)24119887 for any given 119901

1198713119860lt 119901 lt

1199011198673119860

the expected sales volume is given by

119864 [119902 (119901)] = 119911119860(119901) int

119906

1198760(119901)119911119860(119901)

119909119891 (119909) 119889119909

= Γ (119901)

=(119886 minus 119887119901) 119906

2

2 (119906 minus 119897)minus

1198922

2 (119906 minus 119897) (119901 minus ℎ)2

(119886 minus 119887119901)

(A20)

The first and second derivatives of 119864[119902(119901)] wrt 119901 are givenby the below respectively

119889Γ (119901)

119889119901=

[1198922(2119886 + 119887ℎ minus 3119887119901) minus 119887119906

2(119901 minus ℎ)

3

(119886 minus 119887119901)2

]

2 (119906 minus 119897) (119901 minus ℎ)3

(119886 minus 119887119901)2

(A21)

1198892Γ (119901)

1198891199012

=minus1198922

3 (119906 minus 119897) (119901 minus ℎ)4

(119886 minus 119887119901)3

times [ 181198872(119901 minus

(2119886 + 119887ℎ)

3119887)

2

+ (119886 minus 119887ℎ)2]

lt 0 forall119901 gt 0

(A22)

Therefore Γ(119901) is strictly concave in 119901 for all 119901 gt 0Hence there is a unique maximum of Γ(119901) Solving thefirst-order condition by (A21) the unique optimal 119901 thatmaximizes Γ(119901) is given by

1199014119860=arg119901

1198922(2119886 + 119887ℎ minus 3119887119901) minus 119887119906

2(119901 minus ℎ)

3

(119886 minus 119887119901)2

= 0

(A23)

If 1199011198713119860

lt 1199014119860lt 1199011198673119860

then 119901 = 1199014119860

is the optimal solutionof maximizing 119864[119902(119901)] and the corresponding optimalMPQis 1198764119860

lowast= 1198760(1199014) = 119892(119901

4minus ℎ)

Next from the proof of Proposition 7lim119901rarr119901

1198673119860

1198760(119901)119911119860(119901) = lim

119901rarr1199011198713119860

1198760(119901)119911119860(119901) = 119906

or equivalently lim119901rarr119901

1198673119860

120578(119901) = lim119901rarr119901

1198713119860

120578(119901) = 0

(120578(119901) is given by (A16)) Therefore lim119901rarr119901

1198673119860

119864[119902(119901)] =

lim119901rarr119901

1198713119860

119864[119902(119901)] = 0 If 1199014119860

le 1199011198713119860

or 1199014119860

ge 1199011198673119860

then by the strict concavity of Γ(119901) Γ(119901) lt 0 for all1199011198713119860

lt 119901 lt 1199011198673119860

By definition 119864[119902(119901)] ge 0 Therefore119864[119902(119901)] = Γ(119901) and (119876

0(119901) 119901) are not feasible for all 119901 gt 0 in

this case

Proof of Proposition 8 Let 1198701119872(119876) = 119876[ℎ + 119892119876]

119887119886

(a) From Propositions 1 and 2 for any119876 gt 0 the designerwill set the retail price as 119901

1119872

lowast(119876) = 119901

0(119876) = ℎ + 119892119876

Next for (119876 1199010(119876)) being feasible as 119897 lt 119909 lt 119906 we need

to have 1198701119872(119876) lt 119906 Moreover we have 119889119870

1119872(119876)119889119876 =

[ℎ + 119892119876]119887minus1ℎ + 119892(1 minus 119887)119876119886

(b) The expected profit of the retailer is given by

119864 [1205871119872(119876)]

= int

119906

1198701119872(119876)

([1199011119872

lowast

(119876) minus ℎ] 119909119911119872[1199011119872

lowast

(119876)] minus 119892) 119891 (119909) 119889119909

=119892 [119906 minus 119870

1119872(119876)]

(119906 minus 119897)[119886

119876(119892

119876+ ℎ)

minus119887

(119906 + 119870

1119872(119876)

2) minus 1]

(A24)


The first derivative of 119864[1205871119872(119876)] is

119889119864 [1205871119872(119876)]

119889119876

= minus119886119892119876minus3[ℎ +

119892

119876]

minus119887minus1

[1199062minus 1198701119898

2

(119876)]ℎ119876 minus (119887 minus 1) 119892

2 (119906 minus 119897)

(A25)

For 0 lt 119887 le 1 we have ℎ119876 minus (119887 minus 1)119892 ge ℎ119876 gt 0 so119889119864[1205871119872(119876)]119889119876 lt 0 for all 119876 Therefore 119864[120587

1119872(119876)] is

strictly decreasing in 119876 and the optimal MPS should be thesmallest possible value of 119876 For 119876 = 0 119870

1119872(0) = 0 (as

1198701119872(119876) = 119876

1119887ℎ + 119892119876

1119887minus1]119887119886 and 0 lt 119887 le 1) and hence

119864[1205871119872(0)] = +infin

(c) For 119887 gt 1 119889119864[1205871119872(119876)]119889119876 gt 0 for 119876 lt (119887 minus

1)119892ℎ 119889119864[1205871119872(119876)]119889119876 = 0 for 119876 = (119887 minus 1)119892ℎ and

119889119864[1205871119872(119876)]119889119876 lt 0 for 119876 gt (119887 minus 1)119892ℎ Therefore

119864[1205871119872(119876)] is unimodal for 119876 gt 0

(c-i) 1198891198701119872(119876)119889119876 lt 0 for all 119876 lt 119892(119887 minus 1)ℎ

1198891198701119872(119876)119889119876 = 0 for all119876 = 119892(119887minus1)ℎ and 119889119870

1119872(119876)119889119876 gt

0 for all 119876 gt 119892(119887 minus 1)ℎ Therefore 1198701119872(119876) is minimized at

119876 = 119892(119887minus1)ℎ As1198701119872(119892(119887minus1)ℎ) = 119892(119887minus1)[ℎ119887(119887minus1)]

119887119886ℎ

if119892(119887minus1)[ℎ119887(119887minus1)]119887119886ℎ ge 119906 then theRetailerrsquos participationconstraint is not satisfied for any 119876 gt 0 that is 119870

1119872(119876) gt 119906

for all 119876 gt 0(c-ii) If 119892(119887 minus 1)[ℎ119887(119887 minus 1)]119887119886ℎ lt 119906 the optimal MPQ

is uniquely given by 1198761119872

lowast= (119887 minus 1)119892ℎ and by direct

manipulation the optimal retail price is uniquely given by1199011119872

lowast= 1199011119872

lowast(1198761119872

lowast) = 119887ℎ(119887 minus 1)

Proof of Proposition 9 (a) Under the ldquoProject-oriented 119863-ledrdquo scenario with 119911(119901) = 119911

119872(119901) = 119886119901

minus119887 by Proposition 3the optimal MPQ is 119876

0(119901) = 119892(119901 minus ℎ) which satisfies (C1)

Since 119897 le 119909 le 119906 to ensure (1198763119872

lowast(119901) 119901) being feasible we

need to have 1198760(119901)119911119872(119901) lt 119906 which is equivalent to

1198703119872(119901) =

119892119901119887

119886 (119901 minus ℎ)lt 119906 (A26)

(b) and (c)We have 1198891198703119872(119901)119889119901 = (119892119901

119887119886(119901minusℎ)

2)[(119887minus1)119901minus

119887ℎ] Next anticipating 1198760(119901) the DPS feasibility probability

is given by

Ψ3119872(119901) = 1 minus 119865[

1198760(119901)

119911119872(119901)

] (A27)

If 119897 lt 1198703119872(119901) lt 119906

Ψ3119872(119901) = minus

119892

119886 (119906 minus 119897)(

119901119887

119901 minus ℎ) + (1 +

119897

119906 minus 119897) (A28)

and the first derivative of Ψ3119872(119901) wrt 119901 is

119889Ψ3119872(119901)

119889119901= minus

119892119901119887minus1

119886 (119906 minus 119897) (119901 minus ℎ)2[(119887 minus 1) 119901 minus ℎ119887] (A29)

If 119897 ge 1198703119872(119901) Ψ

3119872(119901)= 1 and if 119906 le 119870

3119872(119901) Ψ

3119872(119901) = 0

(note that (1198763119872

lowast(119901) 119901) is not feasible in this case so it can

be ignored in the rest of the proof)

For 119887 le 1 as 1198891198703119872(119901)119889119901 lt 0 for all 119901 gt 0 there exist

unique1199013119872119897

1015840 and unique1199013119872119906

1015840 such that (i)1199013119872119897

1015840gt 1199013119872119906

1015840gt

ℎ (ii) 1198703119872(1199013119872119897

1015840) = 119897 and (iii) 119870

3119872(1199013119872119906

1015840) = 119906 Observed

from (A29) that 119889Ψ3119872(119901)119889119901 gt 0 for all 119901 gt 0 satisfying

119897 lt 1198703119872(119901) lt 119906 Therefore Ψ

3119872(119901) is strictly increasing in 119901

for all 1199013119872119906

1015840lt 119901 lt 119901

3119872119897

1015840 and Ψ3119872(119897) = 1 for all 119901 ge 119901

3119872119897

1015840

((1198760(119901) 119901) is not feasible for any 119901 lt 119901

3119872119906

1015840)For 119887 gt 1 119889119870

3119872(119901)119889119901 lt 0 for all ℎ lt 119901 lt ℎ119887(119887 minus 1)

1198891198703119872(119901)119889119901 = 0 for 119901 = ℎ119887(119887minus1) and 119889119870

3119872(119901)119889119901 gt 0 for

all 119901 gt ℎ119887(119887minus 1) Thus minus1198703119872(119901) is unimodal in 119901 for 119901 gt ℎ

and is maximized at 119901 = 1199013119872

= ℎ119887(119887 minus 1) As 1198703119872(1199013119872) =

(119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))119887 if (119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))119887 ge 119906

then 1198703119872(119901) ge 119906 for all 119901 gt ℎ Therefore the pair (119876

0(119901) 119901)

is feasible if and only if (119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))119887 lt 119906For 119887 gt 1 and (119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))

119887lt 119906 as

1198703119872(ℎ) = +infin there exist 119901

3119872119906

10158401015840 and 1199013119872119906

101584010158401015840 such that (i)ℎ lt 119901

3119872119906

10158401015840lt 1199013119872119906

101584010158401015840 (ii) 1198703119872(1199013119872119906

10158401015840) = 119870

3119872(1199013119872119906

101584010158401015840) = 119906

and (iii) 1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 If(119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))

119887gt 119897 then 119897 lt 119870

3119872(119901) lt 119906 for

all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 Otherwise there exist 1199013119872119897

10158401015840 and1199013119872119897

101584010158401015840 such that (i) 1199013119872119906

10158401015840lt 1199013119872119897

10158401015840le 1199013119872119897

101584010158401015840lt 1199013119872119906

101584010158401015840(ii) 119870

3119872(1199013119872119897

10158401015840) = 119870

3119872(1199013119872119897

101584010158401015840) = 119897 (iii) 119897 lt 119870

3119872(119901) lt 119906

for all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 or 1199013119872119897

101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 and(iv)1198703119872(119901) le 119897 for all 119901

3119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840Next for 119887 gt 1 and 119897 lt 119870

3119872(119901) lt 119906 119889Ψ

3119872(119901)119889119901 gt 0 for

all ℎ lt 119901 lt ℎ119887(119887 minus 1) 119889Ψ3119872(119901)119889119901 = 0 for 119901 = ℎ119887(119887 minus 1)

and 119889Ψ3119872(119901)119889119901 lt 0 for all 119901 gt ℎ119887(119887 minus 1)

Combining the above findings for 119887 gt 1 and (119892(119887 minus1)119886ℎ)(ℎ119887(119887 minus 1))

119887lt 119906 if (119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))119887 gt 119897

then Ψ3119872(119901) is unimodal in 119901 and is maximized at 119901 =

ℎ119887(119887minus1) OtherwiseΨ3119872(119901) is strictly increasing in 119901 for all

1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 is strictly decreasing in119901 for all1199013119872119897

101584010158401015840lt

119901 lt 1199013119872119906

101584010158401015840 and Ψ3119872(119901) = 1 for all 119901

3119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840

Proof of Proposition 10 By Proposition 3(a) anticipating1198760(119901) = 119892(119901 minus ℎ) the expected sales volume is given by

119864 [1199024119872(119901)] = Γ

1(119901) = int

119906

1198703119872(119901)

119909119891 (119909) 119889119909

=1199062

2 (119906 minus 119897)minus

11989221199012119887

21198862(119906 minus 119897) (119901 minus ℎ)

2

(A30)

for 119897 lt 1198703119872(119901) lt 119906 where119870

3119872(119901) is given by (23)

119864 [1199024119872(119901)] =

119906 minus 119897

2 for 119870

3119872(119901) le 119897

119864 [1199024119872(119901)] = 0 for 119870

3119872(119901) ge 119906

(A31)

The first and second derivatives of Γ1(119901) wrt 119901 are derived

below respectively

119889Γ1(119901)

119889119901= minus

11989221199012119887minus1

1198862(119906 minus 119897) (119901 minus ℎ)

3[(119887 minus 1) 119901 minus 119887ℎ] (A32)

For 119887 le 1 observing from (A32) 119889Γ1(119901)119889119901 gt 0 for

all 119901 gt 0 Therefore 119864[1199024119872(119901)] is strictly increasing in 119901

for all 119897 lt 1198703119872(119901) lt 119906 From the proof of Proposition 8


1198891198703119872(119901)119889119901 lt 0 for all 119901 gt 0 for 119887 le 1 Thus 119897 lt 119870

3119872(119901) lt

119906 implies 1199013119872119906

1015840lt 119901 lt 119901

3119872119897

1015840 and 1198703119872(119901) le 119897 implies 119901 ge

1199013119872119897

1015840 where 1198703119872(1199013119872119897

1015840) = 119897 and 119870

3119872(1199013119872119906

1015840) = 119906 (see the

proof of Proposition 8 for details) Moreover 119864[1199024119872(119901)] lt

119864[1199024119872(1199013119872119897

1015840)] = (119906 minus 119897)2 for all 119901

3119872119906

1015840lt 119901 lt 119901

3119872119897

1015840 and119864[1199024119872(119901)] = (119906 minus 119897)2 for all 119901 ge 119901

3119872119897

1015840For 119887 gt 1 and 119897 lt 119870

3119872(119901) lt 119906 119889Γ

1(119901)119889119901 gt 0 for all

ℎ lt 119901 lt ℎ119887(119887 minus 1) 119889Γ1(119901)119889119901 = 0 for 119901 = ℎ119887(119887 minus 1) and

119889Γ1(119901)119889119901 lt 0 for all 119901 gt ℎ119887(119887 minus 1) Next from the proof of

Proposition 8(i) for 119887 gt 1 the pair (119876lowast

0(119901) 119901) is feasible if and only if

(119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))119887lt 119906

(ii) for 119887 gt 1 and (119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))119887lt 119906

1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906

101584010158401015840(iii) for 119887 gt 1 and 119897 lt (119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))

119887lt 119906

119897 lt 1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 and(iv) for 119887 gt 1 and (119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))119887 le 119897 119897 lt

1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 or 1199013119872119897

101584010158401015840lt

119901 lt 1199013119872119906

101584010158401015840 and 1198703119872(119901) le 119897 for all 119901

3119872119897

10158401015840lt 119901 lt

1199013119872119897

101584010158401015840Combining the above findings for 119887 gt 1 and (119892(119887 minus

1)119886ℎ)(ℎ119887(119887 minus 1))119887lt 119906 if (119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))119887 gt 119897

then 119864[1199024119872(119901)] is unimodal in 119901 and is maximized at 119901 =

ℎ119887(119887 minus 1) Otherwise 119864[1199024119872(119901)] is strictly increasing in 119901

for all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 is strictly decreasing in 119901 forall 1199013119872119897

101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 and 119864[1199024119872(119901)] = (119906 minus 119897)2 for all

1199013119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This paper is partially supported by the ldquo985 Projectrdquo ofSun Yat-Sen University The authors sincerely thank theanonymous reviewers for their constructive comments onthis paper

References

[1] M Newbery A Study of the UK Designer Fashion SectorDepartment of Trade amp Industry and the British FashionCouncil London UK 2003

[2] J Richardson ldquoVertical integration and rapid response infashion apparelrdquoOrganization Science vol 7 no 4 pp 400ndash4121996

[3] C Mills ldquoEnterprise orientations a framework for makingsense of fashion sector start-uprdquo International Journal ofEntrepreneurial Behaviour and Research vol 17 no 3 pp 245ndash271 2011

[4] J Howe ldquoThe rise of crowdsourcingrdquo Wired Magazine vol 14no 6 pp 1ndash4 2006

[5] K J Boudreau N Lacetera and K R Lakhani ldquoIncentivesand problem uncertainty in innovation contests an empiricalanalysisrdquoManagement Science vol 57 no 5 pp 843ndash863 2011

[6] P M Di M M Wasko and R E Hooker ldquoGetting customersrsquoideas to work for you learning from dell how to succeed withonline user innovation communitiesrdquoMIS Quarterly Executivevol 9 no 4 pp 213ndash228 2010

[7] Y Huang P V Singh and T Mukhopadhyay ldquoHow to designcrowdsourcing contest a structural empirical analysisrdquo Work-ing Paper 2012

[8] B L Bayus ldquoCrowdsourcing new product ideas over timean analysis of the Dell IdeaStorm communityrdquo ManagementScience vol 59 no 1 pp 226ndash244 2013

[9] YWang ldquoJoint pricing-production decisions in supply chains ofcomplementary products with uncertain demandrdquo OperationsResearch vol 54 no 6 pp 1110ndash1127 2006

[10] A J Dalal and M Alghalith ldquoProduction decisions underjoint price and production uncertaintyrdquo European Journal ofOperational Research vol 197 no 1 pp 84ndash92 2009

[11] B Jiang K Jerath and K Srinivasan ldquoFirm strategies in theldquomid tailrdquo of platform-based retailingrdquo Marketing Science vol30 no 5 pp 757ndash775 2011

[12] Y M Li and Y L Lee ldquoPricing peer-produced servicesquality capacity and competition issuesrdquo European Journal ofOperational Research vol 207 no 3 pp 1658ndash1668 2010

[13] P Li-Liang and T Jiang ldquoStudy on pricing level of C2C elec-tronic commerce platformrdquo in Proceedings of the InternationalConference on Management Science and Engineering (ICMSErsquo11) pp 452ndash455 Rome Italy September 2011

[14] K S Anand and R Aron ldquoGroup buying on the web a compar-ison of price-discovery mechanismsrdquoManagement Science vol49 no 11 pp 1546ndash1562 2003

[15] B A Pasternack ldquoOptimal pricing and return policies forperishable commoditiesrdquo Marketing Science vol 4 no 2 pp166ndash176 1985

[16] G Cai Z G Zhang and M Zhang ldquoGame theoreticalperspectives on dual-channel supply chain competition withprice discounts and pricing schemesrdquo International Journal ofProduction Economics vol 117 no 1 pp 80ndash96 2009

[17] N C Petruzzi and M Dada ldquoPricing and the newsvendorproblem a reviewwith extensionsrdquoOperations Research vol 47no 2 pp 183ndash194 1999

[18] C T Yang L Y Ouyang andHHWu ldquoRetailerrsquos optimal pric-ing and ordering policies for non-instantaneous deterioratingitems with price-dependent demand and partial backloggingrdquoMathematical Problems in Engineering vol 2009 Article ID198305 18 pages 2009

[19] H Chen Y Chen C H Chiu T M Choi and S SethildquoCoordination mechanism for the supply chain with leadtimeconsideration and price-dependent demandrdquo European Journalof Operational Research vol 203 no 1 pp 70ndash80 2010

[20] C H Chiu T M Choi and C S Tang ldquoPrice rebateand returns supply contracts for coordinating supply chainswith price-dependent demandsrdquo Production and OperationsManagement vol 20 no 1 pp 81ndash91 2011

[21] M Fisher and A Raman ldquoReducing the cost of demand uncer-tainty through accurate response to early salesrdquo OperationsResearch vol 44 no 1 pp 87ndash99 1996

[22] S C Graves A H G R Kan and P H Zipkin Handbooks inOperations Research andManagement Science vol 4 of Logisticsof Production and Inventory North Holland Amsterdam TheNetherlands 1993


[23] P S Chow T M Choi and T C E Cheng ldquoImpacts ofminimum order quantity on a quick response supply chainrdquoIEEE Transactions on Systems Man and Cybernetics A vol 42no 4 pp 868ndash879 2012

Submit your manuscripts athttpwwwhindawicom

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Stochastic AnalysisInternational Journal of


also defined the sponsoring crowd as ldquothe new pool of cheaplabor everyday people using their spare cycles to createcontent solve problems and even do corporate RampDrdquo Manyworks have been done recently on crowdsourcing contests[5] and noncompetitive crowdsourcing ideation initiatives[6ndash8] For example the crowdsourcing t-shirt platformteespringcom provides a way for people to create and sell t-shirts online to raisemoneyHere is how it works first peopledesign a t-shirt and post it on the website choose a goal andset a price to launch a campaign page second people sendthe campaign page to supporters and let them preorder the t-shirts towards the campaign goal (preordering is free buyerswill only be charged if the goal is reached) third peoplecan continue to sell shirts past the goal until the campaignends Once it does the platform handle production andshipment people will get a check for the profit The businessmodel of teespringcom which combines the mechanism ofcrowdsourcing and group buying on the web has provideda platform for entrant designers to try their abilities in thereal market practice In this business model there are twokey decisions to make the price of the product and the targetvolume for production (equal to the campaign goal in the caseof teespringcom) which are similar to the widely discussedproblem of ldquojoint pricing-production decisionsrdquo in operationmanagement [9 10]

To better illustrate how this business model workswe consider a simplified case Designer Platform Service(DPS) which works as a retailer connecting designers withcustomers and designers with manufacturers Similarly DPScrowdsources the product design tasks to the entrant designercommunity and allows the customers to make purchasesin the way of group buying First entrant designers postdesign samples or sketches of products on the website ofDPS and then consumers preorder the product (each at afixed price) online based on the product information Theentrant designer decides the selling price of the productwhile the DPS decides the minimum production quantity(MPQ) Only when the preorder quantity reaches or passesthe MPQ threshold will DPS arrange for production anddelivery Preorder customers then make their payment andget the products delivered Otherwise no production willbe arranged and no one will be charged This novel serviceprovides variety to customers boosts the growth of entrantdesigners and links designing works with real markets Inthe whole process we are most interested in how the priceand MPQ decisions are made within DPS with considera-tion of entrant designerrsquos objective decision sequences andcustomer demand structures

Through the study in the form of Stackelberg games andby considering various scenarios we investigate the decisionsin themechanismwith B2C transactions an entrant designerutilizes DPS to promote her design to consumer and wereveal the price-discovery mechanism in this e-market Weconsider that in our model the two key players the entrantdesigner and DSP have very different objectives Due to thespecial stage of an entrant designer in her professional careerher priority mission is to leverage the creativity with marketacceptance so that she can later find out a way to boost herdesign to profitability Our findings provide insights on how

the mechanism protects DPS in pricing decision in a highlyversatile and risky market and how the mechanism providesa feasible path for entrant designers to test and adjust tothe market with limited loss Moreover our findings suggestfeasibility of the DPS business model

The paper contributes in providing insights of equilib-rium decisions in the combined business model of crowd-sourcing and group buyingThe paper is new in investigatinga creative business model which could significantly utilizecreativity and reduce cost The paper is also innovative inconsidering business players having nonmonetary objectivesBy investigating the two key decisions in the DSP we showhow an online platform could raise a business by crowdsourc-ing designs arranging manufacturing and selling to groupbuying customers Specifically we derive optimal decisionsfor the twoparties under the different objectives of the entrantdesigner decision sequences and market demands throughwhich we show how an entrant designer could benefit fromDSP The findings provide guidelines to practitioners inplatform service and designers and help utilize creation withprofit

The remainder of the paper is organized as follows InSection 2 we review the related literature We construct theStackelberg game models in Section 3 and find the bestresponse decisions in Section 4 In Section 5 we derive theequilibrium decisions of the games with specific demandfunctions We discuss and conclude in Section 6 with limi-tations and future research directions

2 Literature Review

This paper relates to the literature (eg [11ndash13]) in serviceplatform and group buying business on the web Web-basedgroup buying mechanisms are being widely used in bothbusiness-to-business (B2B) and business-to-consumer (B2C)transactions Li and Lee [12] show that with the uncertainpeer-produced services quality a monopolistic platformprovider has no incentive in offering multiple quality classesof service while two competing platform providers may offeridentical service contracts but still receive nonnegative profitAnand and Aron [14] studied a monopolist offering Web-based group buying under different kinds of demand uncer-tainty They derive the monopolistrsquos optimal group-buyingschedule under varying conditions of heterogeneity in thedemand regimes and compare its profits with those obtainedunder the more conventional posted-price mechanism

Pricing decision and determination of theminimum pro-duction quantity with price-dependent demand are the mainfocus of the models explored in this paper In marketing andoperations management literature there is a huge amount ofrelated studies under various settings and we review someof them as follows For instance Pasternack [15] studies thepricing decision and return policy for perishable productsCai et al [16] explore from the game-theoretic perspectivethe pricing scheme of a supply chain with dual channels thatcompete with one another The authors show that a simpleprice discount contract can achieve channel coordination Inparticular the pricing issue has aroused the attention ofmany






119910 (119901 119909) = 119911 (119901) 119909 (1)




given by

119862 (119902) = 119892 + ℎ119902 (2)


119888 (119902) =119862 (119902)

119902=

119892

119902 + ℎ (3)

and hence119889119888 (119902)

119889119902= minus119892119902



119902 (119901 119876 119909) = 119910 (119901 119909) if 119910 (119901 119909) ge 1198760 ow

(5)



120587 (119876 119901 119909)

=

120572 [119901119910 (119901 119909) minus 119862 (119910 (119901 119909))] if 119910 (119901 119909) ge 119876

0 ow


(6)


119864 [120587 (119876 119901)] = 120572int

infin

119876119911(119901)

(119901 minus ℎ) 119911 (119901) 119909 minus 119892119891 (119909) 119889119909 (7)


Ψ (119901119876) = Pr119909 ge 119876

119911 (119901) = 1 minus 119865[

119876

119911 (119901)] (8)


119902 (119901 119876 119909) = 119910 (119901 119909) = 119909119911 (119901) if 119909119911 (119901) ge 1198760 ow

(9)


119864 [119902 (119901 119876)] = 119911 (119901)int

infin

119876119911(119901)

119909119891 (119909) 119889119909 (10)






designer


Ψ(119901 119876)

st 120572120587(119876 119901) ge 0 for all 119909 ge 0


119864[120587(119876 119901lowast)]


designer


119864[119902(119901 119876)]

st 120572120587(119876 119901) ge 0 for all 119909 ge 0


119864[120587(119876 119901lowast)]




119864[120587(119876 119901)]


Ψ(119901 119876lowast)





119864[120587(119876 119901)]


119864[119902(119901 119876lowast)]




1199010(119876) = ℎ +

119892

119876(11)








1(119876) = 119901

0(119876)


1


1198761

lowast= arg119876 ge 0 119876119911(ℎ +

119892

119876) + 119892119911

1015840(ℎ +

119892

119876) = 0

(12)




119901lowast

2(119876) = 119901

0(119876) (13)


2


1198762

lowast= arg119876 gt 0 119876119911(ℎ +

119892

119876) + 119892119911

1015840(ℎ +

119892

119876) = 0

(14)








(119901 minus ℎ) 119911 (119901) 119891 [119876

119911 (119901)] + [(119901 minus ℎ)119876 minus 119892]119891

1015840[

119876

119911 (119901)] gt 0

forall119876 gt 0

(15)



3

lowast(119901) = 119892(119901 minus ℎ)


1199013

lowast= arg 119901 gt 119901

0 119911 (119901) + (119901 minus ℎ) 119911

1015840(119901) = 0 (16)




(119901 minus ℎ) 119911 (119901) 119891 [119876

119911 (119901)] + [(119901 minus ℎ)119876 minus 119892]119891

1015840[

119876

119911 (119901)] gt 0

forall119876 gt 0

(17)



1198764

lowast(119901) =

119892

119901 minus ℎ(18)


1199014

lowast= arg 119901 gt 119901

0 120577 (119901) = 0 (19)

where

120577 (119901) =(119901 minus ℎ) 1199111015840(119901) ((119901 minus ℎ)

2

1199112(119901)int

infin

119892[(119901minusℎ)119911(119901)]

119909119891 (119909) 119889119909

+ 1198922119891[

119892

(119901 minus ℎ) 119911 (119901)])

+ 1198922119911 (119901) 119891(

119892

(119901 minus ℎ) 119911 (119901))

(20)


3

lowast(119901) = 119876

4

lowast(119901) = 119892(119901 minus ℎ)


1198760(119901) =

119892

119901 minus ℎ (21)





(1) additive form

119911119860(119901) =

119886 minus 119887119901 for 0 lt 119901 lt 119886119887

0 for 119901 ge 119886119887

(22)


119860(119901) = 0) and







2

1199064119887 2119887119892(119886 minus 119887ℎ) (119886 + 119887ℎ)2119887


2

1199064119887 and 1199011198714119860

lt 1199014119860lt 1199011198674119860

119892(1199014119860minus ℎ) Given by 119901

4119860









1119860

lowast= 2119892119887(119886 minus 119887ℎ) gt


1119860

lowast= (119886 + 119887ℎ)2119887 gt ℎ



1198673119860= (119886 + 119887ℎ)2119887 + radicΔ2119906119887




119887ℎ)24119887



1198763119860

lowast= 2119887119892(119886 minus 119887ℎ)


24119887



4119860= arg1199011198922(2119886+119887ℎminus3119887119901)minus119887119906

2(119901minusℎ)

3(119886minus119887119901)

2=

0 If 1199011198713119860

lt 1199014119860lt 1199011198673119860



lowast= 119892(119901

4119860minus ℎ)









1198714119860lt 1199014119860

lt

1199011198674119860



119911119872(119901) = 119886119901





1119872(119876)] is




minus119887








rium 119901 are given by 1198761119872


1199011119872


Let Δ3119860

= 119906[119906(119886 minus 119887ℎ)2minus 4119887119892] 119901

1198713119860= (119886 + 119887ℎ)2119887 minus

radicΔ31198602119906119887 and 119901

1198673119860= (119886 + 119887ℎ)2119887 +radicΔ

31198602119906119887 Moreover

let

1198703119872(119901) =

119892119901119887

119886 (119901 minus ℎ)lt 119906 (23)

and denote 1199013119872119897

1015840 1199013119872119906

1015840 1199013119872119906

10158401015840 1199013119872119906

101584010158401015840 1199013119872119897

10158401015840 1199013119872119897

1015840101584010158401199013119872119897

1015840 and 1199013119872119906

1015840 that satisfy (i) 1199013119872119897

1015840gt 1199013119872119906

1015840gt ℎ (ii)

1198703119872(1199013119872119897

1015840) = 119897 and (iii) 119870

3119872(1199013119872119906

1015840) = 119906 and 119901

3119872119906

101584010158401199013119872119906

101584010158401015840 1199013119872119897

10158401015840 and 1199013119872119897

101584010158401015840 satisfy (i) ℎ lt 1199013119872119906

10158401015840lt 1199013119872119897

10158401015840le

1199013119872119897

101584010158401015840lt 1199013119872119906

101584010158401015840 (ii) 1198703119872(1199013119872119906

10158401015840) = 119870

3119872(1199013119872119906

101584010158401015840) = 119906 (iii)

1198703119872(1199013119872119897

10158401015840) = 119870

3119872(1199013119872119897

101584010158401015840) = 119897 (iv) 119897 lt 119870

3119872(119901) lt 119906 for

all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 or 1199013119872119897

101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 and (v)1198703119872(119901) le 119897 for all 119901

3119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840




3119872119906

1015840lt 119901 lt 119901

3119872119897

1015840 and Ψ(119897)=1 for all 119901 ge 119901

3119872119897

1015840(c) for 119887 gt 1




3119872119906

10158401015840lt 119901 lt

1199013119872119897


101584010158401015840lt

119901 lt 1199013119872119906

101584010158401015840 andΨ3119872(119901) = 1 for all 119901

3119872119897

10158401015840lt 119901 lt

1199013119872119897

101584010158401015840







1015840lt119901lt119901

3119872119897

1015840 and119864[119902(119901 119876

0(119901))] = (119906 minus 119897)2 for all 119901 ge 119901

3119872119897

1015840(b) for 119887 gt 1


lt 119906then 119864[119902(119901 119876



119887lt 119897 then


1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897


3119872119897

101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 andΨ3119872(119901) = 1 for

all 1199013119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840





With 119911(119901) = 119911119872(119901) = 119886119901


119860(119901) = 119886 minus 119887119901







6 Discussion



Appendix

Mathematical Proofs




119889Ψ (119901)

119889119901= 119891[

119876

119911 (119901)] [

1198761199111015840(119901)

1199112(119901)

] le 0 as 1199111015840 (119901) lt 0

119891 (119909) ge 0 forall119909 ge 0

(A1)


0(119876) by (C1)


lowast(119876) = 119901



119864 [120587 (119876)] =120572int

infin

119876119911[1199012

lowast(119876)]

[(119892

119876)119911 (119901

2

lowast

(119876)) 119909 minus 119892]119891 (119909) 119889119909

= 120572119892[119911 (ℎ + 119892119876)

119876int

infin

119876119911(ℎ+119892119876)

119909119891 (119909) 119889119909

+ 119865(119876

119911 (ℎ + 119892119876)) minus 1]

(A2)

119889119864[120587 (119876)]

119889119876= minus(

120572119892

1198763)[119876119911(ℎ +

119892

119876) + 119892119911

1015840(ℎ +

119892

119876)]

times int

infin

119876119911(ℎ+119892119876)

119909119891 (119909) 119889119909

(A3)


119876119911(ℎ+119892119876)119909119891(119909)119889119909 gt (119876119911(ℎ+


119876=1198762


lowast119911(ℎ + 119892119876

2

lowast) + 119892119911

1015840(ℎ +

1198921198762

lowast) = 0



infin

119876119911(119901)119909119891(119909)119889119909 and

we have 119889119864[119902(119901)]119889119901 = 1199111015840(119901)[intinfin

119876119911(119901)119909119891(119909)119889119909 +

1198762119891[119876119911(119901)]119911

2(119901)] le 0 as 1199111015840(119901) lt 0 and 119891(119909) ge





119864 [120587 (119876)] = 120572int

infin

119876119911(119901)

[(119901 minus ℎ) 119911 (119901) 119909 minus 119892] 119891 (119909) 119889119909

= 120572((119901 minus ℎ) 119911 (119901)int

infin

119876119911(119901)

119909119891 (119909) 119889119909

+119892119865[119876

119911 (119901)] + 119892)

(A4)



119889119864 [120587 (119876)]

119889119876= minus

120572119891 [119876119911 (119901)] [(119901 minus ℎ)119876 minus 119892]

119911 (119901) (A5)

1198892119864 [120587 (119876)]

1198891198762

= minus120572((119901 minus ℎ) 119911 (119901) 119891[119876

119911 (119901)]

+ [(119901 minus ℎ)119876 minus 119892]1198911015840[

119876

119911 (119901)]) times (119911

2(119901))

minus1

(A6)


(119901 minus ℎ) 119911 (119901) 119891 [119876

119911 (119901)] + [(119901 minus ℎ)119876 minus 119892]119891

1015840[

119876

119911 (119901)]

ge (119901 minus ℎ) 119911 (119901) 119891 [119876

119911 (119901)] gt 0

(A7)




3


is given by

Ψ (119901) = 1 minus 119865[1198763

lowast(119901)

119911 (119901)] = 1 minus 119865[

119892

(119901 minus ℎ) 119911 (119901)] (A8)


119889Ψ (119901)

119889119901= [

119892

(119901 minus ℎ)2

1199112(119901)

] [119911 (119901) + (119901 minus ℎ) 1199111015840(119901)]

times 119891[119892

(119901 minus ℎ) 119911 (119901)]

(A9)



119911(1199013

lowast) + (119901

3


1015840(1199013


(A9)



4




is in the form

119889119864 [119902 (119901)]

119889119901=

120577 (119901)

[(119901 minus ℎ)3

1199112(119901)]

(A10)


4


4






1119860(119876) where 119870

1119860(119876) =

119876119911119860(1199011119860


1119860(119876) lt 119906


1198762minus (119886 minus 119887ℎ) 119906119876 + 119887119892119906

gt 1198762minus (119886 minus 119887ℎ) 119906119876 +

(119886 minus 119887ℎ)21199062

4

= [119876 minus(119886 minus 119887ℎ) 119906

2]

2

ge 0

(A11)


given by

1205871119860(119876)

=

[1199011119860

lowast(119876) minus ℎ] 119909119911

119860[1199011119860


1119860(119876)

0 ow(A12)

where 1199011119860

lowast(119876) = 119901



119864 [1205871119860(119876)]

= int

119906

1198701119860(119876)

([1199011119860

lowast

(119876) minus ℎ] 119909119911119860[1199011119860

lowast

(119876)] minus 119892) 119891 (119909) 119889119909

=119892 [119906 minus 119870

1119860(119876)]

(119906 minus 119897)([119886 minus 119887ℎ

119876minus119887119892

1198762] [

119906 + 1198701119860(119876)

2] minus 1)

(A13)



119889119864 [1205871119860(119876)]

119889119876=

119892 [1199062minus 1198701119860

2(119876)] [minus (119886 minus 119887ℎ)119876 + 2119887119892]

2 (119906 minus 119897) 1198763

(A14)

1198892119864 [1205871119860(119876)]

1198891198762

=

119892 [1199062minus 1198701119860

2(119876)] [(119886 minus 119887ℎ)119876 minus 3119887119892]

(119906 minus 119897) 1198764

(A15)

From (A15) for 119876 gt 0 1198892119864[1205871119860(119876)]119889119876


3119887119892(119886 minus 119887ℎ) (as 1199062 gt 1198701119860

2(119876)) Thus 119864[120587

1119860(119876)] is strictly





1119860(119876)]119889119876 = 0) Therefore 119864[120587

1119860(119876)]


lowast= 2119887119892(119886 minus 119887ℎ) is


lowast= 1199011119860

lowast(1198761119860

lowast) =

(119886 + 119887ℎ)2119887


119860(119901) = 119886 minus 119887119901





120578 (119901) = 1199061198871199012minus 119906 (119886 + 119887ℎ) 119901 + 119906ℎ119886 + 119892 lt 0 (A16)


lt 119901 lt

1199011198673119860


Ψ3(119901) = 1 minus 119865[

1198763119860

lowast(119901)

119911119860(119901)

]

= minus119892


119906

(119906 minus 119897)]

(A17)



119889Ψ (119901)

119889119901=

119892

(119906 minus 119897) (119901 minus ℎ)2

(119886 minus 119887119901)2(119886 + ℎ119887 minus 2119887119901) (A18)

1198892Ψ (119901)

1198891199012

= minus2119892

(119906 minus 119897) (119901 minus ℎ)3

(119886 minus 119887119901)3


] lt 0

(A19)

for all 1199011198713119860

lt 119901 lt 1199011198673119860


for 1199011198713119860

lt 119901 lt 1199011198673119860


3119860

lowast= (119886 + ℎ119887)2119887


lt 1199013119860

lowastlt 1199011198673119860


3119860

lowast=

1198760(1199013119860

lowast) = 2119887119892(119886 minus 119887ℎ)


119860(119901) = 119886 minus 119887119901


1198713119860lt 119901 lt

1199011198673119860


119864 [119902 (119901)] = 119911119860(119901) int

119906

1198760(119901)119911119860(119901)

119909119891 (119909) 119889119909

= Γ (119901)

=(119886 minus 119887119901) 119906

2

2 (119906 minus 119897)minus

1198922

2 (119906 minus 119897) (119901 minus ℎ)2

(119886 minus 119887119901)

(A20)


119889Γ (119901)

119889119901=

[1198922(2119886 + 119887ℎ minus 3119887119901) minus 119887119906

2(119901 minus ℎ)

3

(119886 minus 119887119901)2

]

2 (119906 minus 119897) (119901 minus ℎ)3

(119886 minus 119887119901)2

(A21)

1198892Γ (119901)

1198891199012

=minus1198922

3 (119906 minus 119897) (119901 minus ℎ)4

(119886 minus 119887119901)3

times [ 181198872(119901 minus

(2119886 + 119887ℎ)

3119887)

2

+ (119886 minus 119887ℎ)2]


(A22)


1199014119860=arg119901

1198922(2119886 + 119887ℎ minus 3119887119901) minus 119887119906

2(119901 minus ℎ)

3

(119886 minus 119887119901)2

= 0

(A23)

If 1199011198713119860

lt 1199014119860lt 1199011198673119860

then 119901 = 1199014119860


lowast= 1198760(1199014) = 119892(119901

4minus ℎ)


1198673119860

1198760(119901)119911119860(119901) = lim

119901rarr1199011198713119860

1198760(119901)119911119860(119901) = 119906


1198673119860

120578(119901) = lim119901rarr119901

1198713119860

120578(119901) = 0


1198673119860

119864[119902(119901)] =

lim119901rarr119901

1198713119860

119864[119902(119901)] = 0 If 1199014119860

le 1199011198713119860

or 1199014119860

ge 1199011198673119860


lt 119901 lt 1199011198673119860



this case


119887119886


1119872

lowast(119876) = 119901

0(119876) = ℎ + 119892119876



1119872(119876)119889119876 =

[ℎ + 119892119876]119887minus1ℎ + 119892(1 minus 119887)119876119886


119864 [1205871119872(119876)]

= int

119906

1198701119872(119876)

([1199011119872

lowast

(119876) minus ℎ] 119909119911119872[1199011119872

lowast

(119876)] minus 119892) 119891 (119909) 119889119909

=119892 [119906 minus 119870

1119872(119876)]

(119906 minus 119897)[119886

119876(119892

119876+ ℎ)

minus119887

(119906 + 119870

1119872(119876)

2) minus 1]

(A24)



119889119864 [1205871119872(119876)]

119889119876

= minus119886119892119876minus3[ℎ +

119892

119876]

minus119887minus1

[1199062minus 1198701119898

2

(119876)]ℎ119876 minus (119887 minus 1) 119892

2 (119906 minus 119897)

(A25)


1119872(119876)] is


1119872(0) = 0 (as

1198701119872(119876) = 119876

1119887ℎ + 119892119876


119864[1205871119872(0)] = +infin


1)119892ℎ 119889119864[1205871119872(119876)]119889119876 = 0 for 119876 = (119887 minus 1)119892ℎ and





1119872(119876)119889119876 gt



119887119886ℎ


1119872(119876) gt 119906





lowast= 1199011119872

lowast(1198761119872

lowast) = 119887ℎ(119887 minus 1)


119872(119901) = 119886119901






1198703119872(119901) =

119892119901119887

119886 (119901 minus ℎ)lt 119906 (A26)

(b) and (c)We have 1198891198703119872(119901)119889119901 = (119892119901

119887119886(119901minusℎ)

2)[(119887minus1)119901minus


is given by

Ψ3119872(119901) = 1 minus 119865[

1198760(119901)

119911119872(119901)

] (A27)

If 119897 lt 1198703119872(119901) lt 119906

Ψ3119872(119901) = minus

119892

119886 (119906 minus 119897)(

119901119887

119901 minus ℎ) + (1 +

119897

119906 minus 119897) (A28)


119889Ψ3119872(119901)

119889119901= minus

119892119901119887minus1


If 119897 ge 1198703119872(119901) Ψ

3119872(119901)= 1 and if 119906 le 119870

3119872(119901) Ψ

3119872(119901) = 0

(note that (1198763119872




unique1199013119872119897

1015840 and unique1199013119872119906

1015840 such that (i)1199013119872119897

1015840gt 1199013119872119906

1015840gt

ℎ (ii) 1198703119872(1199013119872119897

1015840) = 119897 and (iii) 119870

3119872(1199013119872119906

1015840) = 119906 Observed


119897 lt 1198703119872(119901) lt 119906 Therefore Ψ


for all 1199013119872119906

1015840lt 119901 lt 119901

3119872119897

1015840 and Ψ3119872(119897) = 1 for all 119901 ge 119901

3119872119897

1015840


3119872119906

1015840)For 119887 gt 1 119889119870


1198891198703119872(119901)119889119901 = 0 for 119901 = ℎ119887(119887minus1) and 119889119870

3119872(119901)119889119901 gt 0 for



= ℎ119887(119887 minus 1) As 1198703119872(1199013119872) =



0(119901) 119901)


119887lt 119906 as


3119872119906

10158401015840 and 1199013119872119906

101584010158401015840 such that (i)ℎ lt 119901

3119872119906

10158401015840lt 1199013119872119906

101584010158401015840 (ii) 1198703119872(1199013119872119906

10158401015840) = 119870

3119872(1199013119872119906

101584010158401015840) = 119906

and (iii) 1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 If(119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))

119887gt 119897 then 119897 lt 119870

3119872(119901) lt 119906 for

all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119906


10158401015840 and1199013119872119897

101584010158401015840 such that (i) 1199013119872119906

10158401015840lt 1199013119872119897

10158401015840le 1199013119872119897

101584010158401015840lt 1199013119872119906

101584010158401015840(ii) 119870

3119872(1199013119872119897

10158401015840) = 119870

3119872(1199013119872119897

101584010158401015840) = 119897 (iii) 119897 lt 119870

3119872(119901) lt 119906

for all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 or 1199013119872119897

101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 and(iv)1198703119872(119901) le 119897 for all 119901

3119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840Next for 119887 gt 1 and 119897 lt 119870

3119872(119901) lt 119906 119889Ψ

3119872(119901)119889119901 gt 0 for







1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897


101584010158401015840lt

119901 lt 1199013119872119906

101584010158401015840 and Ψ3119872(119901) = 1 for all 119901

3119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840


119864 [1199024119872(119901)] = Γ

1(119901) = int

119906

1198703119872(119901)

119909119891 (119909) 119889119909

=1199062

2 (119906 minus 119897)minus

11989221199012119887

21198862(119906 minus 119897) (119901 minus ℎ)

2

(A30)

for 119897 lt 1198703119872(119901) lt 119906 where119870

3119872(119901) is given by (23)

119864 [1199024119872(119901)] =

119906 minus 119897

2 for 119870

3119872(119901) le 119897

119864 [1199024119872(119901)] = 0 for 119870

3119872(119901) ge 119906

(A31)


below respectively

119889Γ1(119901)

119889119901= minus

11989221199012119887minus1

1198862(119906 minus 119897) (119901 minus ℎ)

3[(119887 minus 1) 119901 minus 119887ℎ] (A32)






3119872(119901) lt

119906 implies 1199013119872119906

1015840lt 119901 lt 119901

3119872119897

1015840 and 1198703119872(119901) le 119897 implies 119901 ge

1199013119872119897

1015840 where 1198703119872(1199013119872119897

1015840) = 119897 and 119870

3119872(1199013119872119906

1015840) = 119906 (see the


119864[1199024119872(1199013119872119897

1015840)] = (119906 minus 119897)2 for all 119901

3119872119906

1015840lt 119901 lt 119901

3119872119897


3119872119897

1015840For 119887 gt 1 and 119897 lt 119870

3119872(119901) lt 119906 119889Γ

1(119901)119889119901 gt 0 for all





(119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))119887lt 119906


1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906


119887lt 119906

119897 lt 1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906


1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 or 1199013119872119897

101584010158401015840lt

119901 lt 1199013119872119906

101584010158401015840 and 1198703119872(119901) le 119897 for all 119901

3119872119897

10158401015840lt 119901 lt

1199013119872119897





for all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897


101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 and 119864[1199024119872(119901)] = (119906 minus 119897)2 for all

1199013119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840



Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119910 (119901 119909) = 119911 (119901) 119909 (1)




given by

119862 (119902) = 119892 + ℎ119902 (2)


119888 (119902) =119862 (119902)

119902=

119892

119902 + ℎ (3)

and hence119889119888 (119902)

119889119902= minus119892119902



119902 (119901 119876 119909) = 119910 (119901 119909) if 119910 (119901 119909) ge 1198760 ow

(5)



120587 (119876 119901 119909)

=

120572 [119901119910 (119901 119909) minus 119862 (119910 (119901 119909))] if 119910 (119901 119909) ge 119876

0 ow


(6)


119864 [120587 (119876 119901)] = 120572int

infin

119876119911(119901)

(119901 minus ℎ) 119911 (119901) 119909 minus 119892119891 (119909) 119889119909 (7)


Ψ (119901119876) = Pr119909 ge 119876

119911 (119901) = 1 minus 119865[

119876

119911 (119901)] (8)


119902 (119901 119876 119909) = 119910 (119901 119909) = 119909119911 (119901) if 119909119911 (119901) ge 1198760 ow

(9)


119864 [119902 (119901 119876)] = 119911 (119901)int

infin

119876119911(119901)

119909119891 (119909) 119889119909 (10)






designer


Ψ(119901 119876)

st 120572120587(119876 119901) ge 0 for all 119909 ge 0


119864[120587(119876 119901lowast)]


designer


119864[119902(119901 119876)]

st 120572120587(119876 119901) ge 0 for all 119909 ge 0


119864[120587(119876 119901lowast)]




119864[120587(119876 119901)]


Ψ(119901 119876lowast)





119864[120587(119876 119901)]


119864[119902(119901 119876lowast)]




1199010(119876) = ℎ +

119892

119876(11)








1(119876) = 119901

0(119876)


1


1198761

lowast= arg119876 ge 0 119876119911(ℎ +

119892

119876) + 119892119911

1015840(ℎ +

119892

119876) = 0

(12)




119901lowast

2(119876) = 119901

0(119876) (13)


2


1198762

lowast= arg119876 gt 0 119876119911(ℎ +

119892

119876) + 119892119911

1015840(ℎ +

119892

119876) = 0

(14)








(119901 minus ℎ) 119911 (119901) 119891 [119876

119911 (119901)] + [(119901 minus ℎ)119876 minus 119892]119891

1015840[

119876

119911 (119901)] gt 0

forall119876 gt 0

(15)



3

lowast(119901) = 119892(119901 minus ℎ)


1199013

lowast= arg 119901 gt 119901

0 119911 (119901) + (119901 minus ℎ) 119911

1015840(119901) = 0 (16)




(119901 minus ℎ) 119911 (119901) 119891 [119876

119911 (119901)] + [(119901 minus ℎ)119876 minus 119892]119891

1015840[

119876

119911 (119901)] gt 0

forall119876 gt 0

(17)



1198764

lowast(119901) =

119892

119901 minus ℎ(18)


1199014

lowast= arg 119901 gt 119901

0 120577 (119901) = 0 (19)

where

120577 (119901) =(119901 minus ℎ) 1199111015840(119901) ((119901 minus ℎ)

2

1199112(119901)int

infin

119892[(119901minusℎ)119911(119901)]

119909119891 (119909) 119889119909

+ 1198922119891[

119892

(119901 minus ℎ) 119911 (119901)])

+ 1198922119911 (119901) 119891(

119892

(119901 minus ℎ) 119911 (119901))

(20)


3

lowast(119901) = 119876

4

lowast(119901) = 119892(119901 minus ℎ)


1198760(119901) =

119892

119901 minus ℎ (21)





(1) additive form

119911119860(119901) =

119886 minus 119887119901 for 0 lt 119901 lt 119886119887

0 for 119901 ge 119886119887

(22)


119860(119901) = 0) and







2

1199064119887 2119887119892(119886 minus 119887ℎ) (119886 + 119887ℎ)2119887


2

1199064119887 and 1199011198714119860

lt 1199014119860lt 1199011198674119860

119892(1199014119860minus ℎ) Given by 119901

4119860









1119860

lowast= 2119892119887(119886 minus 119887ℎ) gt


1119860

lowast= (119886 + 119887ℎ)2119887 gt ℎ



1198673119860= (119886 + 119887ℎ)2119887 + radicΔ2119906119887




119887ℎ)24119887



1198763119860

lowast= 2119887119892(119886 minus 119887ℎ)


24119887



4119860= arg1199011198922(2119886+119887ℎminus3119887119901)minus119887119906

2(119901minusℎ)

3(119886minus119887119901)

2=

0 If 1199011198713119860

lt 1199014119860lt 1199011198673119860



lowast= 119892(119901

4119860minus ℎ)









1198714119860lt 1199014119860

lt

1199011198674119860



119911119872(119901) = 119886119901





1119872(119876)] is




minus119887








rium 119901 are given by 1198761119872


1199011119872


Let Δ3119860

= 119906[119906(119886 minus 119887ℎ)2minus 4119887119892] 119901

1198713119860= (119886 + 119887ℎ)2119887 minus

radicΔ31198602119906119887 and 119901

1198673119860= (119886 + 119887ℎ)2119887 +radicΔ

31198602119906119887 Moreover

let

1198703119872(119901) =

119892119901119887

119886 (119901 minus ℎ)lt 119906 (23)

and denote 1199013119872119897

1015840 1199013119872119906

1015840 1199013119872119906

10158401015840 1199013119872119906

101584010158401015840 1199013119872119897

10158401015840 1199013119872119897

1015840101584010158401199013119872119897

1015840 and 1199013119872119906

1015840 that satisfy (i) 1199013119872119897

1015840gt 1199013119872119906

1015840gt ℎ (ii)

1198703119872(1199013119872119897

1015840) = 119897 and (iii) 119870

3119872(1199013119872119906

1015840) = 119906 and 119901

3119872119906

101584010158401199013119872119906

101584010158401015840 1199013119872119897

10158401015840 and 1199013119872119897

101584010158401015840 satisfy (i) ℎ lt 1199013119872119906

10158401015840lt 1199013119872119897

10158401015840le

1199013119872119897

101584010158401015840lt 1199013119872119906

101584010158401015840 (ii) 1198703119872(1199013119872119906

10158401015840) = 119870

3119872(1199013119872119906

101584010158401015840) = 119906 (iii)

1198703119872(1199013119872119897

10158401015840) = 119870

3119872(1199013119872119897

101584010158401015840) = 119897 (iv) 119897 lt 119870

3119872(119901) lt 119906 for

all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 or 1199013119872119897

101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 and (v)1198703119872(119901) le 119897 for all 119901

3119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840




3119872119906

1015840lt 119901 lt 119901

3119872119897

1015840 and Ψ(119897)=1 for all 119901 ge 119901

3119872119897

1015840(c) for 119887 gt 1




3119872119906

10158401015840lt 119901 lt

1199013119872119897


101584010158401015840lt

119901 lt 1199013119872119906

101584010158401015840 andΨ3119872(119901) = 1 for all 119901

3119872119897

10158401015840lt 119901 lt

1199013119872119897

101584010158401015840







1015840lt119901lt119901

3119872119897

1015840 and119864[119902(119901 119876

0(119901))] = (119906 minus 119897)2 for all 119901 ge 119901

3119872119897

1015840(b) for 119887 gt 1


lt 119906then 119864[119902(119901 119876



119887lt 119897 then


1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897


3119872119897

101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 andΨ3119872(119901) = 1 for

all 1199013119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840





With 119911(119901) = 119911119872(119901) = 119886119901


119860(119901) = 119886 minus 119887119901







6 Discussion



Appendix

Mathematical Proofs




119889Ψ (119901)

119889119901= 119891[

119876

119911 (119901)] [

1198761199111015840(119901)

1199112(119901)

] le 0 as 1199111015840 (119901) lt 0

119891 (119909) ge 0 forall119909 ge 0

(A1)


0(119876) by (C1)


lowast(119876) = 119901



119864 [120587 (119876)] =120572int

infin

119876119911[1199012

lowast(119876)]

[(119892

119876)119911 (119901

2

lowast

(119876)) 119909 minus 119892]119891 (119909) 119889119909

= 120572119892[119911 (ℎ + 119892119876)

119876int

infin

119876119911(ℎ+119892119876)

119909119891 (119909) 119889119909

+ 119865(119876

119911 (ℎ + 119892119876)) minus 1]

(A2)

119889119864[120587 (119876)]

119889119876= minus(

120572119892

1198763)[119876119911(ℎ +

119892

119876) + 119892119911

1015840(ℎ +

119892

119876)]

times int

infin

119876119911(ℎ+119892119876)

119909119891 (119909) 119889119909

(A3)


119876119911(ℎ+119892119876)119909119891(119909)119889119909 gt (119876119911(ℎ+


119876=1198762


lowast119911(ℎ + 119892119876

2

lowast) + 119892119911

1015840(ℎ +

1198921198762

lowast) = 0



infin

119876119911(119901)119909119891(119909)119889119909 and

we have 119889119864[119902(119901)]119889119901 = 1199111015840(119901)[intinfin

119876119911(119901)119909119891(119909)119889119909 +

1198762119891[119876119911(119901)]119911

2(119901)] le 0 as 1199111015840(119901) lt 0 and 119891(119909) ge





119864 [120587 (119876)] = 120572int

infin

119876119911(119901)

[(119901 minus ℎ) 119911 (119901) 119909 minus 119892] 119891 (119909) 119889119909

= 120572((119901 minus ℎ) 119911 (119901)int

infin

119876119911(119901)

119909119891 (119909) 119889119909

+119892119865[119876

119911 (119901)] + 119892)

(A4)



119889119864 [120587 (119876)]

119889119876= minus

120572119891 [119876119911 (119901)] [(119901 minus ℎ)119876 minus 119892]

119911 (119901) (A5)

1198892119864 [120587 (119876)]

1198891198762

= minus120572((119901 minus ℎ) 119911 (119901) 119891[119876

119911 (119901)]

+ [(119901 minus ℎ)119876 minus 119892]1198911015840[

119876

119911 (119901)]) times (119911

2(119901))

minus1

(A6)


(119901 minus ℎ) 119911 (119901) 119891 [119876

119911 (119901)] + [(119901 minus ℎ)119876 minus 119892]119891

1015840[

119876

119911 (119901)]

ge (119901 minus ℎ) 119911 (119901) 119891 [119876

119911 (119901)] gt 0

(A7)




3


is given by

Ψ (119901) = 1 minus 119865[1198763

lowast(119901)

119911 (119901)] = 1 minus 119865[

119892

(119901 minus ℎ) 119911 (119901)] (A8)


119889Ψ (119901)

119889119901= [

119892

(119901 minus ℎ)2

1199112(119901)

] [119911 (119901) + (119901 minus ℎ) 1199111015840(119901)]

times 119891[119892

(119901 minus ℎ) 119911 (119901)]

(A9)



119911(1199013

lowast) + (119901

3


1015840(1199013


(A9)



4




is in the form

119889119864 [119902 (119901)]

119889119901=

120577 (119901)

[(119901 minus ℎ)3

1199112(119901)]

(A10)


4


4






1119860(119876) where 119870

1119860(119876) =

119876119911119860(1199011119860


1119860(119876) lt 119906


1198762minus (119886 minus 119887ℎ) 119906119876 + 119887119892119906

gt 1198762minus (119886 minus 119887ℎ) 119906119876 +

(119886 minus 119887ℎ)21199062

4

= [119876 minus(119886 minus 119887ℎ) 119906

2]

2

ge 0

(A11)


given by

1205871119860(119876)

=

[1199011119860

lowast(119876) minus ℎ] 119909119911

119860[1199011119860


1119860(119876)

0 ow(A12)

where 1199011119860

lowast(119876) = 119901



119864 [1205871119860(119876)]

= int

119906

1198701119860(119876)

([1199011119860

lowast

(119876) minus ℎ] 119909119911119860[1199011119860

lowast

(119876)] minus 119892) 119891 (119909) 119889119909

=119892 [119906 minus 119870

1119860(119876)]

(119906 minus 119897)([119886 minus 119887ℎ

119876minus119887119892

1198762] [

119906 + 1198701119860(119876)

2] minus 1)

(A13)



119889119864 [1205871119860(119876)]

119889119876=

119892 [1199062minus 1198701119860

2(119876)] [minus (119886 minus 119887ℎ)119876 + 2119887119892]

2 (119906 minus 119897) 1198763

(A14)

1198892119864 [1205871119860(119876)]

1198891198762

=

119892 [1199062minus 1198701119860

2(119876)] [(119886 minus 119887ℎ)119876 minus 3119887119892]

(119906 minus 119897) 1198764

(A15)

From (A15) for 119876 gt 0 1198892119864[1205871119860(119876)]119889119876


3119887119892(119886 minus 119887ℎ) (as 1199062 gt 1198701119860

2(119876)) Thus 119864[120587

1119860(119876)] is strictly





1119860(119876)]119889119876 = 0) Therefore 119864[120587

1119860(119876)]


lowast= 2119887119892(119886 minus 119887ℎ) is


lowast= 1199011119860

lowast(1198761119860

lowast) =

(119886 + 119887ℎ)2119887


119860(119901) = 119886 minus 119887119901





120578 (119901) = 1199061198871199012minus 119906 (119886 + 119887ℎ) 119901 + 119906ℎ119886 + 119892 lt 0 (A16)


lt 119901 lt

1199011198673119860


Ψ3(119901) = 1 minus 119865[

1198763119860

lowast(119901)

119911119860(119901)

]

= minus119892


119906

(119906 minus 119897)]

(A17)



119889Ψ (119901)

119889119901=

119892

(119906 minus 119897) (119901 minus ℎ)2

(119886 minus 119887119901)2(119886 + ℎ119887 minus 2119887119901) (A18)

1198892Ψ (119901)

1198891199012

= minus2119892

(119906 minus 119897) (119901 minus ℎ)3

(119886 minus 119887119901)3


] lt 0

(A19)

for all 1199011198713119860

lt 119901 lt 1199011198673119860


for 1199011198713119860

lt 119901 lt 1199011198673119860


3119860

lowast= (119886 + ℎ119887)2119887


lt 1199013119860

lowastlt 1199011198673119860


3119860

lowast=

1198760(1199013119860

lowast) = 2119887119892(119886 minus 119887ℎ)


119860(119901) = 119886 minus 119887119901


1198713119860lt 119901 lt

1199011198673119860


119864 [119902 (119901)] = 119911119860(119901) int

119906

1198760(119901)119911119860(119901)

119909119891 (119909) 119889119909

= Γ (119901)

=(119886 minus 119887119901) 119906

2

2 (119906 minus 119897)minus

1198922

2 (119906 minus 119897) (119901 minus ℎ)2

(119886 minus 119887119901)

(A20)


119889Γ (119901)

119889119901=

[1198922(2119886 + 119887ℎ minus 3119887119901) minus 119887119906

2(119901 minus ℎ)

3

(119886 minus 119887119901)2

]

2 (119906 minus 119897) (119901 minus ℎ)3

(119886 minus 119887119901)2

(A21)

1198892Γ (119901)

1198891199012

=minus1198922

3 (119906 minus 119897) (119901 minus ℎ)4

(119886 minus 119887119901)3

times [ 181198872(119901 minus

(2119886 + 119887ℎ)

3119887)

2

+ (119886 minus 119887ℎ)2]


(A22)


1199014119860=arg119901

1198922(2119886 + 119887ℎ minus 3119887119901) minus 119887119906

2(119901 minus ℎ)

3

(119886 minus 119887119901)2

= 0

(A23)

If 1199011198713119860

lt 1199014119860lt 1199011198673119860

then 119901 = 1199014119860


lowast= 1198760(1199014) = 119892(119901

4minus ℎ)


1198673119860

1198760(119901)119911119860(119901) = lim

119901rarr1199011198713119860

1198760(119901)119911119860(119901) = 119906


1198673119860

120578(119901) = lim119901rarr119901

1198713119860

120578(119901) = 0


1198673119860

119864[119902(119901)] =

lim119901rarr119901

1198713119860

119864[119902(119901)] = 0 If 1199014119860

le 1199011198713119860

or 1199014119860

ge 1199011198673119860


lt 119901 lt 1199011198673119860



this case


119887119886


1119872

lowast(119876) = 119901

0(119876) = ℎ + 119892119876



1119872(119876)119889119876 =

[ℎ + 119892119876]119887minus1ℎ + 119892(1 minus 119887)119876119886


119864 [1205871119872(119876)]

= int

119906

1198701119872(119876)

([1199011119872

lowast

(119876) minus ℎ] 119909119911119872[1199011119872

lowast

(119876)] minus 119892) 119891 (119909) 119889119909

=119892 [119906 minus 119870

1119872(119876)]

(119906 minus 119897)[119886

119876(119892

119876+ ℎ)

minus119887

(119906 + 119870

1119872(119876)

2) minus 1]

(A24)



119889119864 [1205871119872(119876)]

119889119876

= minus119886119892119876minus3[ℎ +

119892

119876]

minus119887minus1

[1199062minus 1198701119898

2

(119876)]ℎ119876 minus (119887 minus 1) 119892

2 (119906 minus 119897)

(A25)


1119872(119876)] is


1119872(0) = 0 (as

1198701119872(119876) = 119876

1119887ℎ + 119892119876


119864[1205871119872(0)] = +infin


1)119892ℎ 119889119864[1205871119872(119876)]119889119876 = 0 for 119876 = (119887 minus 1)119892ℎ and





1119872(119876)119889119876 gt



119887119886ℎ


1119872(119876) gt 119906





lowast= 1199011119872

lowast(1198761119872

lowast) = 119887ℎ(119887 minus 1)


119872(119901) = 119886119901






1198703119872(119901) =

119892119901119887

119886 (119901 minus ℎ)lt 119906 (A26)

(b) and (c)We have 1198891198703119872(119901)119889119901 = (119892119901

119887119886(119901minusℎ)

2)[(119887minus1)119901minus


is given by

Ψ3119872(119901) = 1 minus 119865[

1198760(119901)

119911119872(119901)

] (A27)

If 119897 lt 1198703119872(119901) lt 119906

Ψ3119872(119901) = minus

119892

119886 (119906 minus 119897)(

119901119887

119901 minus ℎ) + (1 +

119897

119906 minus 119897) (A28)


119889Ψ3119872(119901)

119889119901= minus

119892119901119887minus1


If 119897 ge 1198703119872(119901) Ψ

3119872(119901)= 1 and if 119906 le 119870

3119872(119901) Ψ

3119872(119901) = 0

(note that (1198763119872




unique1199013119872119897

1015840 and unique1199013119872119906

1015840 such that (i)1199013119872119897

1015840gt 1199013119872119906

1015840gt

ℎ (ii) 1198703119872(1199013119872119897

1015840) = 119897 and (iii) 119870

3119872(1199013119872119906

1015840) = 119906 Observed


119897 lt 1198703119872(119901) lt 119906 Therefore Ψ


for all 1199013119872119906

1015840lt 119901 lt 119901

3119872119897

1015840 and Ψ3119872(119897) = 1 for all 119901 ge 119901

3119872119897

1015840


3119872119906

1015840)For 119887 gt 1 119889119870


1198891198703119872(119901)119889119901 = 0 for 119901 = ℎ119887(119887minus1) and 119889119870

3119872(119901)119889119901 gt 0 for



= ℎ119887(119887 minus 1) As 1198703119872(1199013119872) =



0(119901) 119901)


119887lt 119906 as


3119872119906

10158401015840 and 1199013119872119906

101584010158401015840 such that (i)ℎ lt 119901

3119872119906

10158401015840lt 1199013119872119906

101584010158401015840 (ii) 1198703119872(1199013119872119906

10158401015840) = 119870

3119872(1199013119872119906

101584010158401015840) = 119906

and (iii) 1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 If(119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))

119887gt 119897 then 119897 lt 119870

3119872(119901) lt 119906 for

all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119906


10158401015840 and1199013119872119897

101584010158401015840 such that (i) 1199013119872119906

10158401015840lt 1199013119872119897

10158401015840le 1199013119872119897

101584010158401015840lt 1199013119872119906

101584010158401015840(ii) 119870

3119872(1199013119872119897

10158401015840) = 119870

3119872(1199013119872119897

101584010158401015840) = 119897 (iii) 119897 lt 119870

3119872(119901) lt 119906

for all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 or 1199013119872119897

101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 and(iv)1198703119872(119901) le 119897 for all 119901

3119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840Next for 119887 gt 1 and 119897 lt 119870

3119872(119901) lt 119906 119889Ψ

3119872(119901)119889119901 gt 0 for







1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897


101584010158401015840lt

119901 lt 1199013119872119906

101584010158401015840 and Ψ3119872(119901) = 1 for all 119901

3119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840


119864 [1199024119872(119901)] = Γ

1(119901) = int

119906

1198703119872(119901)

119909119891 (119909) 119889119909

=1199062

2 (119906 minus 119897)minus

11989221199012119887

21198862(119906 minus 119897) (119901 minus ℎ)

2

(A30)

for 119897 lt 1198703119872(119901) lt 119906 where119870

3119872(119901) is given by (23)

119864 [1199024119872(119901)] =

119906 minus 119897

2 for 119870

3119872(119901) le 119897

119864 [1199024119872(119901)] = 0 for 119870

3119872(119901) ge 119906

(A31)


below respectively

119889Γ1(119901)

119889119901= minus

11989221199012119887minus1

1198862(119906 minus 119897) (119901 minus ℎ)

3[(119887 minus 1) 119901 minus 119887ℎ] (A32)






3119872(119901) lt

119906 implies 1199013119872119906

1015840lt 119901 lt 119901

3119872119897

1015840 and 1198703119872(119901) le 119897 implies 119901 ge

1199013119872119897

1015840 where 1198703119872(1199013119872119897

1015840) = 119897 and 119870

3119872(1199013119872119906

1015840) = 119906 (see the


119864[1199024119872(1199013119872119897

1015840)] = (119906 minus 119897)2 for all 119901

3119872119906

1015840lt 119901 lt 119901

3119872119897


3119872119897

1015840For 119887 gt 1 and 119897 lt 119870

3119872(119901) lt 119906 119889Γ

1(119901)119889119901 gt 0 for all





(119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))119887lt 119906


1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906


119887lt 119906

119897 lt 1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906


1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 or 1199013119872119897

101584010158401015840lt

119901 lt 1199013119872119906

101584010158401015840 and 1198703119872(119901) le 119897 for all 119901

3119872119897

10158401015840lt 119901 lt

1199013119872119897





for all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897


101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 and 119864[1199024119872(119901)] = (119906 minus 119897)2 for all

1199013119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840



Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













designer


Ψ(119901 119876)

st 120572120587(119876 119901) ge 0 for all 119909 ge 0


119864[120587(119876 119901lowast)]


designer


119864[119902(119901 119876)]

st 120572120587(119876 119901) ge 0 for all 119909 ge 0


119864[120587(119876 119901lowast)]




119864[120587(119876 119901)]


Ψ(119901 119876lowast)





119864[120587(119876 119901)]


119864[119902(119901 119876lowast)]




1199010(119876) = ℎ +

119892

119876(11)








1(119876) = 119901

0(119876)


1


1198761

lowast= arg119876 ge 0 119876119911(ℎ +

119892

119876) + 119892119911

1015840(ℎ +

119892

119876) = 0

(12)




119901lowast

2(119876) = 119901

0(119876) (13)


2


1198762

lowast= arg119876 gt 0 119876119911(ℎ +

119892

119876) + 119892119911

1015840(ℎ +

119892

119876) = 0

(14)








(119901 minus ℎ) 119911 (119901) 119891 [119876

119911 (119901)] + [(119901 minus ℎ)119876 minus 119892]119891

1015840[

119876

119911 (119901)] gt 0

forall119876 gt 0

(15)



3

lowast(119901) = 119892(119901 minus ℎ)


1199013

lowast= arg 119901 gt 119901

0 119911 (119901) + (119901 minus ℎ) 119911

1015840(119901) = 0 (16)




(119901 minus ℎ) 119911 (119901) 119891 [119876

119911 (119901)] + [(119901 minus ℎ)119876 minus 119892]119891

1015840[

119876

119911 (119901)] gt 0

forall119876 gt 0

(17)



1198764

lowast(119901) =

119892

119901 minus ℎ(18)


1199014

lowast= arg 119901 gt 119901

0 120577 (119901) = 0 (19)

where

120577 (119901) =(119901 minus ℎ) 1199111015840(119901) ((119901 minus ℎ)

2

1199112(119901)int

infin

119892[(119901minusℎ)119911(119901)]

119909119891 (119909) 119889119909

+ 1198922119891[

119892

(119901 minus ℎ) 119911 (119901)])

+ 1198922119911 (119901) 119891(

119892

(119901 minus ℎ) 119911 (119901))

(20)


3

lowast(119901) = 119876

4

lowast(119901) = 119892(119901 minus ℎ)


1198760(119901) =

119892

119901 minus ℎ (21)





(1) additive form

119911119860(119901) =

119886 minus 119887119901 for 0 lt 119901 lt 119886119887

0 for 119901 ge 119886119887

(22)


119860(119901) = 0) and







2

1199064119887 2119887119892(119886 minus 119887ℎ) (119886 + 119887ℎ)2119887


2

1199064119887 and 1199011198714119860

lt 1199014119860lt 1199011198674119860

119892(1199014119860minus ℎ) Given by 119901

4119860









1119860

lowast= 2119892119887(119886 minus 119887ℎ) gt


1119860

lowast= (119886 + 119887ℎ)2119887 gt ℎ



1198673119860= (119886 + 119887ℎ)2119887 + radicΔ2119906119887




119887ℎ)24119887



1198763119860

lowast= 2119887119892(119886 minus 119887ℎ)


24119887



4119860= arg1199011198922(2119886+119887ℎminus3119887119901)minus119887119906

2(119901minusℎ)

3(119886minus119887119901)

2=

0 If 1199011198713119860

lt 1199014119860lt 1199011198673119860



lowast= 119892(119901

4119860minus ℎ)









1198714119860lt 1199014119860

lt

1199011198674119860



119911119872(119901) = 119886119901





1119872(119876)] is




minus119887








rium 119901 are given by 1198761119872


1199011119872


Let Δ3119860

= 119906[119906(119886 minus 119887ℎ)2minus 4119887119892] 119901

1198713119860= (119886 + 119887ℎ)2119887 minus

radicΔ31198602119906119887 and 119901

1198673119860= (119886 + 119887ℎ)2119887 +radicΔ

31198602119906119887 Moreover

let

1198703119872(119901) =

119892119901119887

119886 (119901 minus ℎ)lt 119906 (23)

and denote 1199013119872119897

1015840 1199013119872119906

1015840 1199013119872119906

10158401015840 1199013119872119906

101584010158401015840 1199013119872119897

10158401015840 1199013119872119897

1015840101584010158401199013119872119897

1015840 and 1199013119872119906

1015840 that satisfy (i) 1199013119872119897

1015840gt 1199013119872119906

1015840gt ℎ (ii)

1198703119872(1199013119872119897

1015840) = 119897 and (iii) 119870

3119872(1199013119872119906

1015840) = 119906 and 119901

3119872119906

101584010158401199013119872119906

101584010158401015840 1199013119872119897

10158401015840 and 1199013119872119897

101584010158401015840 satisfy (i) ℎ lt 1199013119872119906

10158401015840lt 1199013119872119897

10158401015840le

1199013119872119897

101584010158401015840lt 1199013119872119906

101584010158401015840 (ii) 1198703119872(1199013119872119906

10158401015840) = 119870

3119872(1199013119872119906

101584010158401015840) = 119906 (iii)

1198703119872(1199013119872119897

10158401015840) = 119870

3119872(1199013119872119897

101584010158401015840) = 119897 (iv) 119897 lt 119870

3119872(119901) lt 119906 for

all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 or 1199013119872119897

101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 and (v)1198703119872(119901) le 119897 for all 119901

3119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840




3119872119906

1015840lt 119901 lt 119901

3119872119897

1015840 and Ψ(119897)=1 for all 119901 ge 119901

3119872119897

1015840(c) for 119887 gt 1




3119872119906

10158401015840lt 119901 lt

1199013119872119897


101584010158401015840lt

119901 lt 1199013119872119906

101584010158401015840 andΨ3119872(119901) = 1 for all 119901

3119872119897

10158401015840lt 119901 lt

1199013119872119897

101584010158401015840







1015840lt119901lt119901

3119872119897

1015840 and119864[119902(119901 119876

0(119901))] = (119906 minus 119897)2 for all 119901 ge 119901

3119872119897

1015840(b) for 119887 gt 1


lt 119906then 119864[119902(119901 119876



119887lt 119897 then


1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897


3119872119897

101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 andΨ3119872(119901) = 1 for

all 1199013119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840





With 119911(119901) = 119911119872(119901) = 119886119901


119860(119901) = 119886 minus 119887119901







6 Discussion



Appendix

Mathematical Proofs




119889Ψ (119901)

119889119901= 119891[

119876

119911 (119901)] [

1198761199111015840(119901)

1199112(119901)

] le 0 as 1199111015840 (119901) lt 0

119891 (119909) ge 0 forall119909 ge 0

(A1)


0(119876) by (C1)


lowast(119876) = 119901



119864 [120587 (119876)] =120572int

infin

119876119911[1199012

lowast(119876)]

[(119892

119876)119911 (119901

2

lowast

(119876)) 119909 minus 119892]119891 (119909) 119889119909

= 120572119892[119911 (ℎ + 119892119876)

119876int

infin

119876119911(ℎ+119892119876)

119909119891 (119909) 119889119909

+ 119865(119876

119911 (ℎ + 119892119876)) minus 1]

(A2)

119889119864[120587 (119876)]

119889119876= minus(

120572119892

1198763)[119876119911(ℎ +

119892

119876) + 119892119911

1015840(ℎ +

119892

119876)]

times int

infin

119876119911(ℎ+119892119876)

119909119891 (119909) 119889119909

(A3)


119876119911(ℎ+119892119876)119909119891(119909)119889119909 gt (119876119911(ℎ+


119876=1198762


lowast119911(ℎ + 119892119876

2

lowast) + 119892119911

1015840(ℎ +

1198921198762

lowast) = 0



infin

119876119911(119901)119909119891(119909)119889119909 and

we have 119889119864[119902(119901)]119889119901 = 1199111015840(119901)[intinfin

119876119911(119901)119909119891(119909)119889119909 +

1198762119891[119876119911(119901)]119911

2(119901)] le 0 as 1199111015840(119901) lt 0 and 119891(119909) ge





119864 [120587 (119876)] = 120572int

infin

119876119911(119901)

[(119901 minus ℎ) 119911 (119901) 119909 minus 119892] 119891 (119909) 119889119909

= 120572((119901 minus ℎ) 119911 (119901)int

infin

119876119911(119901)

119909119891 (119909) 119889119909

+119892119865[119876

119911 (119901)] + 119892)

(A4)



119889119864 [120587 (119876)]

119889119876= minus

120572119891 [119876119911 (119901)] [(119901 minus ℎ)119876 minus 119892]

119911 (119901) (A5)

1198892119864 [120587 (119876)]

1198891198762

= minus120572((119901 minus ℎ) 119911 (119901) 119891[119876

119911 (119901)]

+ [(119901 minus ℎ)119876 minus 119892]1198911015840[

119876

119911 (119901)]) times (119911

2(119901))

minus1

(A6)


(119901 minus ℎ) 119911 (119901) 119891 [119876

119911 (119901)] + [(119901 minus ℎ)119876 minus 119892]119891

1015840[

119876

119911 (119901)]

ge (119901 minus ℎ) 119911 (119901) 119891 [119876

119911 (119901)] gt 0

(A7)




3


is given by

Ψ (119901) = 1 minus 119865[1198763

lowast(119901)

119911 (119901)] = 1 minus 119865[

119892

(119901 minus ℎ) 119911 (119901)] (A8)


119889Ψ (119901)

119889119901= [

119892

(119901 minus ℎ)2

1199112(119901)

] [119911 (119901) + (119901 minus ℎ) 1199111015840(119901)]

times 119891[119892

(119901 minus ℎ) 119911 (119901)]

(A9)



119911(1199013

lowast) + (119901

3


1015840(1199013


(A9)



4




is in the form

119889119864 [119902 (119901)]

119889119901=

120577 (119901)

[(119901 minus ℎ)3

1199112(119901)]

(A10)


4


4






1119860(119876) where 119870

1119860(119876) =

119876119911119860(1199011119860


1119860(119876) lt 119906


1198762minus (119886 minus 119887ℎ) 119906119876 + 119887119892119906

gt 1198762minus (119886 minus 119887ℎ) 119906119876 +

(119886 minus 119887ℎ)21199062

4

= [119876 minus(119886 minus 119887ℎ) 119906

2]

2

ge 0

(A11)


given by

1205871119860(119876)

=

[1199011119860

lowast(119876) minus ℎ] 119909119911

119860[1199011119860


1119860(119876)

0 ow(A12)

where 1199011119860

lowast(119876) = 119901



119864 [1205871119860(119876)]

= int

119906

1198701119860(119876)

([1199011119860

lowast

(119876) minus ℎ] 119909119911119860[1199011119860

lowast

(119876)] minus 119892) 119891 (119909) 119889119909

=119892 [119906 minus 119870

1119860(119876)]

(119906 minus 119897)([119886 minus 119887ℎ

119876minus119887119892

1198762] [

119906 + 1198701119860(119876)

2] minus 1)

(A13)



119889119864 [1205871119860(119876)]

119889119876=

119892 [1199062minus 1198701119860

2(119876)] [minus (119886 minus 119887ℎ)119876 + 2119887119892]

2 (119906 minus 119897) 1198763

(A14)

1198892119864 [1205871119860(119876)]

1198891198762

=

119892 [1199062minus 1198701119860

2(119876)] [(119886 minus 119887ℎ)119876 minus 3119887119892]

(119906 minus 119897) 1198764

(A15)

From (A15) for 119876 gt 0 1198892119864[1205871119860(119876)]119889119876


3119887119892(119886 minus 119887ℎ) (as 1199062 gt 1198701119860

2(119876)) Thus 119864[120587

1119860(119876)] is strictly





1119860(119876)]119889119876 = 0) Therefore 119864[120587

1119860(119876)]


lowast= 2119887119892(119886 minus 119887ℎ) is


lowast= 1199011119860

lowast(1198761119860

lowast) =

(119886 + 119887ℎ)2119887


119860(119901) = 119886 minus 119887119901





120578 (119901) = 1199061198871199012minus 119906 (119886 + 119887ℎ) 119901 + 119906ℎ119886 + 119892 lt 0 (A16)


lt 119901 lt

1199011198673119860


Ψ3(119901) = 1 minus 119865[

1198763119860

lowast(119901)

119911119860(119901)

]

= minus119892


119906

(119906 minus 119897)]

(A17)



119889Ψ (119901)

119889119901=

119892

(119906 minus 119897) (119901 minus ℎ)2

(119886 minus 119887119901)2(119886 + ℎ119887 minus 2119887119901) (A18)

1198892Ψ (119901)

1198891199012

= minus2119892

(119906 minus 119897) (119901 minus ℎ)3

(119886 minus 119887119901)3


] lt 0

(A19)

for all 1199011198713119860

lt 119901 lt 1199011198673119860


for 1199011198713119860

lt 119901 lt 1199011198673119860


3119860

lowast= (119886 + ℎ119887)2119887


lt 1199013119860

lowastlt 1199011198673119860


3119860

lowast=

1198760(1199013119860

lowast) = 2119887119892(119886 minus 119887ℎ)


119860(119901) = 119886 minus 119887119901


1198713119860lt 119901 lt

1199011198673119860


119864 [119902 (119901)] = 119911119860(119901) int

119906

1198760(119901)119911119860(119901)

119909119891 (119909) 119889119909

= Γ (119901)

=(119886 minus 119887119901) 119906

2

2 (119906 minus 119897)minus

1198922

2 (119906 minus 119897) (119901 minus ℎ)2

(119886 minus 119887119901)

(A20)


119889Γ (119901)

119889119901=

[1198922(2119886 + 119887ℎ minus 3119887119901) minus 119887119906

2(119901 minus ℎ)

3

(119886 minus 119887119901)2

]

2 (119906 minus 119897) (119901 minus ℎ)3

(119886 minus 119887119901)2

(A21)

1198892Γ (119901)

1198891199012

=minus1198922

3 (119906 minus 119897) (119901 minus ℎ)4

(119886 minus 119887119901)3

times [ 181198872(119901 minus

(2119886 + 119887ℎ)

3119887)

2

+ (119886 minus 119887ℎ)2]


(A22)


1199014119860=arg119901

1198922(2119886 + 119887ℎ minus 3119887119901) minus 119887119906

2(119901 minus ℎ)

3

(119886 minus 119887119901)2

= 0

(A23)

If 1199011198713119860

lt 1199014119860lt 1199011198673119860

then 119901 = 1199014119860


lowast= 1198760(1199014) = 119892(119901

4minus ℎ)


1198673119860

1198760(119901)119911119860(119901) = lim

119901rarr1199011198713119860

1198760(119901)119911119860(119901) = 119906


1198673119860

120578(119901) = lim119901rarr119901

1198713119860

120578(119901) = 0


1198673119860

119864[119902(119901)] =

lim119901rarr119901

1198713119860

119864[119902(119901)] = 0 If 1199014119860

le 1199011198713119860

or 1199014119860

ge 1199011198673119860


lt 119901 lt 1199011198673119860



this case


119887119886


1119872

lowast(119876) = 119901

0(119876) = ℎ + 119892119876



1119872(119876)119889119876 =

[ℎ + 119892119876]119887minus1ℎ + 119892(1 minus 119887)119876119886


119864 [1205871119872(119876)]

= int

119906

1198701119872(119876)

([1199011119872

lowast

(119876) minus ℎ] 119909119911119872[1199011119872

lowast

(119876)] minus 119892) 119891 (119909) 119889119909

=119892 [119906 minus 119870

1119872(119876)]

(119906 minus 119897)[119886

119876(119892

119876+ ℎ)

minus119887

(119906 + 119870

1119872(119876)

2) minus 1]

(A24)



119889119864 [1205871119872(119876)]

119889119876

= minus119886119892119876minus3[ℎ +

119892

119876]

minus119887minus1

[1199062minus 1198701119898

2

(119876)]ℎ119876 minus (119887 minus 1) 119892

2 (119906 minus 119897)

(A25)


1119872(119876)] is


1119872(0) = 0 (as

1198701119872(119876) = 119876

1119887ℎ + 119892119876


119864[1205871119872(0)] = +infin


1)119892ℎ 119889119864[1205871119872(119876)]119889119876 = 0 for 119876 = (119887 minus 1)119892ℎ and





1119872(119876)119889119876 gt



119887119886ℎ


1119872(119876) gt 119906





lowast= 1199011119872

lowast(1198761119872

lowast) = 119887ℎ(119887 minus 1)


119872(119901) = 119886119901






1198703119872(119901) =

119892119901119887

119886 (119901 minus ℎ)lt 119906 (A26)

(b) and (c)We have 1198891198703119872(119901)119889119901 = (119892119901

119887119886(119901minusℎ)

2)[(119887minus1)119901minus


is given by

Ψ3119872(119901) = 1 minus 119865[

1198760(119901)

119911119872(119901)

] (A27)

If 119897 lt 1198703119872(119901) lt 119906

Ψ3119872(119901) = minus

119892

119886 (119906 minus 119897)(

119901119887

119901 minus ℎ) + (1 +

119897

119906 minus 119897) (A28)


119889Ψ3119872(119901)

119889119901= minus

119892119901119887minus1


If 119897 ge 1198703119872(119901) Ψ

3119872(119901)= 1 and if 119906 le 119870

3119872(119901) Ψ

3119872(119901) = 0

(note that (1198763119872




unique1199013119872119897

1015840 and unique1199013119872119906

1015840 such that (i)1199013119872119897

1015840gt 1199013119872119906

1015840gt

ℎ (ii) 1198703119872(1199013119872119897

1015840) = 119897 and (iii) 119870

3119872(1199013119872119906

1015840) = 119906 Observed


119897 lt 1198703119872(119901) lt 119906 Therefore Ψ


for all 1199013119872119906

1015840lt 119901 lt 119901

3119872119897

1015840 and Ψ3119872(119897) = 1 for all 119901 ge 119901

3119872119897

1015840


3119872119906

1015840)For 119887 gt 1 119889119870


1198891198703119872(119901)119889119901 = 0 for 119901 = ℎ119887(119887minus1) and 119889119870

3119872(119901)119889119901 gt 0 for



= ℎ119887(119887 minus 1) As 1198703119872(1199013119872) =



0(119901) 119901)


119887lt 119906 as


3119872119906

10158401015840 and 1199013119872119906

101584010158401015840 such that (i)ℎ lt 119901

3119872119906

10158401015840lt 1199013119872119906

101584010158401015840 (ii) 1198703119872(1199013119872119906

10158401015840) = 119870

3119872(1199013119872119906

101584010158401015840) = 119906

and (iii) 1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 If(119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))

119887gt 119897 then 119897 lt 119870

3119872(119901) lt 119906 for

all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119906


10158401015840 and1199013119872119897

101584010158401015840 such that (i) 1199013119872119906

10158401015840lt 1199013119872119897

10158401015840le 1199013119872119897

101584010158401015840lt 1199013119872119906

101584010158401015840(ii) 119870

3119872(1199013119872119897

10158401015840) = 119870

3119872(1199013119872119897

101584010158401015840) = 119897 (iii) 119897 lt 119870

3119872(119901) lt 119906

for all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 or 1199013119872119897

101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 and(iv)1198703119872(119901) le 119897 for all 119901

3119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840Next for 119887 gt 1 and 119897 lt 119870

3119872(119901) lt 119906 119889Ψ

3119872(119901)119889119901 gt 0 for







1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897


101584010158401015840lt

119901 lt 1199013119872119906

101584010158401015840 and Ψ3119872(119901) = 1 for all 119901

3119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840


119864 [1199024119872(119901)] = Γ

1(119901) = int

119906

1198703119872(119901)

119909119891 (119909) 119889119909

=1199062

2 (119906 minus 119897)minus

11989221199012119887

21198862(119906 minus 119897) (119901 minus ℎ)

2

(A30)

for 119897 lt 1198703119872(119901) lt 119906 where119870

3119872(119901) is given by (23)

119864 [1199024119872(119901)] =

119906 minus 119897

2 for 119870

3119872(119901) le 119897

119864 [1199024119872(119901)] = 0 for 119870

3119872(119901) ge 119906

(A31)


below respectively

119889Γ1(119901)

119889119901= minus

11989221199012119887minus1

1198862(119906 minus 119897) (119901 minus ℎ)

3[(119887 minus 1) 119901 minus 119887ℎ] (A32)






3119872(119901) lt

119906 implies 1199013119872119906

1015840lt 119901 lt 119901

3119872119897

1015840 and 1198703119872(119901) le 119897 implies 119901 ge

1199013119872119897

1015840 where 1198703119872(1199013119872119897

1015840) = 119897 and 119870

3119872(1199013119872119906

1015840) = 119906 (see the


119864[1199024119872(1199013119872119897

1015840)] = (119906 minus 119897)2 for all 119901

3119872119906

1015840lt 119901 lt 119901

3119872119897


3119872119897

1015840For 119887 gt 1 and 119897 lt 119870

3119872(119901) lt 119906 119889Γ

1(119901)119889119901 gt 0 for all





(119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))119887lt 119906


1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906


119887lt 119906

119897 lt 1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906


1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 or 1199013119872119897

101584010158401015840lt

119901 lt 1199013119872119906

101584010158401015840 and 1198703119872(119901) le 119897 for all 119901

3119872119897

10158401015840lt 119901 lt

1199013119872119897





for all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897


101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 and 119864[1199024119872(119901)] = (119906 minus 119897)2 for all

1199013119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840



Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















(119901 minus ℎ) 119911 (119901) 119891 [119876

119911 (119901)] + [(119901 minus ℎ)119876 minus 119892]119891

1015840[

119876

119911 (119901)] gt 0

forall119876 gt 0

(15)



3

lowast(119901) = 119892(119901 minus ℎ)


1199013

lowast= arg 119901 gt 119901

0 119911 (119901) + (119901 minus ℎ) 119911

1015840(119901) = 0 (16)




(119901 minus ℎ) 119911 (119901) 119891 [119876

119911 (119901)] + [(119901 minus ℎ)119876 minus 119892]119891

1015840[

119876

119911 (119901)] gt 0

forall119876 gt 0

(17)



1198764

lowast(119901) =

119892

119901 minus ℎ(18)


1199014

lowast= arg 119901 gt 119901

0 120577 (119901) = 0 (19)

where

120577 (119901) =(119901 minus ℎ) 1199111015840(119901) ((119901 minus ℎ)

2

1199112(119901)int

infin

119892[(119901minusℎ)119911(119901)]

119909119891 (119909) 119889119909

+ 1198922119891[

119892

(119901 minus ℎ) 119911 (119901)])

+ 1198922119911 (119901) 119891(

119892

(119901 minus ℎ) 119911 (119901))

(20)


3

lowast(119901) = 119876

4

lowast(119901) = 119892(119901 minus ℎ)


1198760(119901) =

119892

119901 minus ℎ (21)





(1) additive form

119911119860(119901) =

119886 minus 119887119901 for 0 lt 119901 lt 119886119887

0 for 119901 ge 119886119887

(22)


119860(119901) = 0) and







2

1199064119887 2119887119892(119886 minus 119887ℎ) (119886 + 119887ℎ)2119887


2

1199064119887 and 1199011198714119860

lt 1199014119860lt 1199011198674119860

119892(1199014119860minus ℎ) Given by 119901

4119860









1119860

lowast= 2119892119887(119886 minus 119887ℎ) gt


1119860

lowast= (119886 + 119887ℎ)2119887 gt ℎ



1198673119860= (119886 + 119887ℎ)2119887 + radicΔ2119906119887




119887ℎ)24119887



1198763119860

lowast= 2119887119892(119886 minus 119887ℎ)


24119887



4119860= arg1199011198922(2119886+119887ℎminus3119887119901)minus119887119906

2(119901minusℎ)

3(119886minus119887119901)

2=

0 If 1199011198713119860

lt 1199014119860lt 1199011198673119860



lowast= 119892(119901

4119860minus ℎ)









1198714119860lt 1199014119860

lt

1199011198674119860



119911119872(119901) = 119886119901





1119872(119876)] is




minus119887








rium 119901 are given by 1198761119872


1199011119872


Let Δ3119860

= 119906[119906(119886 minus 119887ℎ)2minus 4119887119892] 119901

1198713119860= (119886 + 119887ℎ)2119887 minus

radicΔ31198602119906119887 and 119901

1198673119860= (119886 + 119887ℎ)2119887 +radicΔ

31198602119906119887 Moreover

let

1198703119872(119901) =

119892119901119887

119886 (119901 minus ℎ)lt 119906 (23)

and denote 1199013119872119897

1015840 1199013119872119906

1015840 1199013119872119906

10158401015840 1199013119872119906

101584010158401015840 1199013119872119897

10158401015840 1199013119872119897

1015840101584010158401199013119872119897

1015840 and 1199013119872119906

1015840 that satisfy (i) 1199013119872119897

1015840gt 1199013119872119906

1015840gt ℎ (ii)

1198703119872(1199013119872119897

1015840) = 119897 and (iii) 119870

3119872(1199013119872119906

1015840) = 119906 and 119901

3119872119906

101584010158401199013119872119906

101584010158401015840 1199013119872119897

10158401015840 and 1199013119872119897

101584010158401015840 satisfy (i) ℎ lt 1199013119872119906

10158401015840lt 1199013119872119897

10158401015840le

1199013119872119897

101584010158401015840lt 1199013119872119906

101584010158401015840 (ii) 1198703119872(1199013119872119906

10158401015840) = 119870

3119872(1199013119872119906

101584010158401015840) = 119906 (iii)

1198703119872(1199013119872119897

10158401015840) = 119870

3119872(1199013119872119897

101584010158401015840) = 119897 (iv) 119897 lt 119870

3119872(119901) lt 119906 for

all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 or 1199013119872119897

101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 and (v)1198703119872(119901) le 119897 for all 119901

3119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840




3119872119906

1015840lt 119901 lt 119901

3119872119897

1015840 and Ψ(119897)=1 for all 119901 ge 119901

3119872119897

1015840(c) for 119887 gt 1




3119872119906

10158401015840lt 119901 lt

1199013119872119897


101584010158401015840lt

119901 lt 1199013119872119906

101584010158401015840 andΨ3119872(119901) = 1 for all 119901

3119872119897

10158401015840lt 119901 lt

1199013119872119897

101584010158401015840







1015840lt119901lt119901

3119872119897

1015840 and119864[119902(119901 119876

0(119901))] = (119906 minus 119897)2 for all 119901 ge 119901

3119872119897

1015840(b) for 119887 gt 1


lt 119906then 119864[119902(119901 119876



119887lt 119897 then


1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897


3119872119897

101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 andΨ3119872(119901) = 1 for

all 1199013119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840





With 119911(119901) = 119911119872(119901) = 119886119901


119860(119901) = 119886 minus 119887119901







6 Discussion



Appendix

Mathematical Proofs




119889Ψ (119901)

119889119901= 119891[

119876

119911 (119901)] [

1198761199111015840(119901)

1199112(119901)

] le 0 as 1199111015840 (119901) lt 0

119891 (119909) ge 0 forall119909 ge 0

(A1)


0(119876) by (C1)


lowast(119876) = 119901



119864 [120587 (119876)] =120572int

infin

119876119911[1199012

lowast(119876)]

[(119892

119876)119911 (119901

2

lowast

(119876)) 119909 minus 119892]119891 (119909) 119889119909

= 120572119892[119911 (ℎ + 119892119876)

119876int

infin

119876119911(ℎ+119892119876)

119909119891 (119909) 119889119909

+ 119865(119876

119911 (ℎ + 119892119876)) minus 1]

(A2)

119889119864[120587 (119876)]

119889119876= minus(

120572119892

1198763)[119876119911(ℎ +

119892

119876) + 119892119911

1015840(ℎ +

119892

119876)]

times int

infin

119876119911(ℎ+119892119876)

119909119891 (119909) 119889119909

(A3)


119876119911(ℎ+119892119876)119909119891(119909)119889119909 gt (119876119911(ℎ+


119876=1198762


lowast119911(ℎ + 119892119876

2

lowast) + 119892119911

1015840(ℎ +

1198921198762

lowast) = 0



infin

119876119911(119901)119909119891(119909)119889119909 and

we have 119889119864[119902(119901)]119889119901 = 1199111015840(119901)[intinfin

119876119911(119901)119909119891(119909)119889119909 +

1198762119891[119876119911(119901)]119911

2(119901)] le 0 as 1199111015840(119901) lt 0 and 119891(119909) ge





119864 [120587 (119876)] = 120572int

infin

119876119911(119901)

[(119901 minus ℎ) 119911 (119901) 119909 minus 119892] 119891 (119909) 119889119909

= 120572((119901 minus ℎ) 119911 (119901)int

infin

119876119911(119901)

119909119891 (119909) 119889119909

+119892119865[119876

119911 (119901)] + 119892)

(A4)



119889119864 [120587 (119876)]

119889119876= minus

120572119891 [119876119911 (119901)] [(119901 minus ℎ)119876 minus 119892]

119911 (119901) (A5)

1198892119864 [120587 (119876)]

1198891198762

= minus120572((119901 minus ℎ) 119911 (119901) 119891[119876

119911 (119901)]

+ [(119901 minus ℎ)119876 minus 119892]1198911015840[

119876

119911 (119901)]) times (119911

2(119901))

minus1

(A6)


(119901 minus ℎ) 119911 (119901) 119891 [119876

119911 (119901)] + [(119901 minus ℎ)119876 minus 119892]119891

1015840[

119876

119911 (119901)]

ge (119901 minus ℎ) 119911 (119901) 119891 [119876

119911 (119901)] gt 0

(A7)




3


is given by

Ψ (119901) = 1 minus 119865[1198763

lowast(119901)

119911 (119901)] = 1 minus 119865[

119892

(119901 minus ℎ) 119911 (119901)] (A8)


119889Ψ (119901)

119889119901= [

119892

(119901 minus ℎ)2

1199112(119901)

] [119911 (119901) + (119901 minus ℎ) 1199111015840(119901)]

times 119891[119892

(119901 minus ℎ) 119911 (119901)]

(A9)



119911(1199013

lowast) + (119901

3


1015840(1199013


(A9)



4




is in the form

119889119864 [119902 (119901)]

119889119901=

120577 (119901)

[(119901 minus ℎ)3

1199112(119901)]

(A10)


4


4






1119860(119876) where 119870

1119860(119876) =

119876119911119860(1199011119860


1119860(119876) lt 119906


1198762minus (119886 minus 119887ℎ) 119906119876 + 119887119892119906

gt 1198762minus (119886 minus 119887ℎ) 119906119876 +

(119886 minus 119887ℎ)21199062

4

= [119876 minus(119886 minus 119887ℎ) 119906

2]

2

ge 0

(A11)


given by

1205871119860(119876)

=

[1199011119860

lowast(119876) minus ℎ] 119909119911

119860[1199011119860


1119860(119876)

0 ow(A12)

where 1199011119860

lowast(119876) = 119901



119864 [1205871119860(119876)]

= int

119906

1198701119860(119876)

([1199011119860

lowast

(119876) minus ℎ] 119909119911119860[1199011119860

lowast

(119876)] minus 119892) 119891 (119909) 119889119909

=119892 [119906 minus 119870

1119860(119876)]

(119906 minus 119897)([119886 minus 119887ℎ

119876minus119887119892

1198762] [

119906 + 1198701119860(119876)

2] minus 1)

(A13)



119889119864 [1205871119860(119876)]

119889119876=

119892 [1199062minus 1198701119860

2(119876)] [minus (119886 minus 119887ℎ)119876 + 2119887119892]

2 (119906 minus 119897) 1198763

(A14)

1198892119864 [1205871119860(119876)]

1198891198762

=

119892 [1199062minus 1198701119860

2(119876)] [(119886 minus 119887ℎ)119876 minus 3119887119892]

(119906 minus 119897) 1198764

(A15)

From (A15) for 119876 gt 0 1198892119864[1205871119860(119876)]119889119876


3119887119892(119886 minus 119887ℎ) (as 1199062 gt 1198701119860

2(119876)) Thus 119864[120587

1119860(119876)] is strictly





1119860(119876)]119889119876 = 0) Therefore 119864[120587

1119860(119876)]


lowast= 2119887119892(119886 minus 119887ℎ) is


lowast= 1199011119860

lowast(1198761119860

lowast) =

(119886 + 119887ℎ)2119887


119860(119901) = 119886 minus 119887119901





120578 (119901) = 1199061198871199012minus 119906 (119886 + 119887ℎ) 119901 + 119906ℎ119886 + 119892 lt 0 (A16)


lt 119901 lt

1199011198673119860


Ψ3(119901) = 1 minus 119865[

1198763119860

lowast(119901)

119911119860(119901)

]

= minus119892


119906

(119906 minus 119897)]

(A17)



119889Ψ (119901)

119889119901=

119892

(119906 minus 119897) (119901 minus ℎ)2

(119886 minus 119887119901)2(119886 + ℎ119887 minus 2119887119901) (A18)

1198892Ψ (119901)

1198891199012

= minus2119892

(119906 minus 119897) (119901 minus ℎ)3

(119886 minus 119887119901)3


] lt 0

(A19)

for all 1199011198713119860

lt 119901 lt 1199011198673119860


for 1199011198713119860

lt 119901 lt 1199011198673119860


3119860

lowast= (119886 + ℎ119887)2119887


lt 1199013119860

lowastlt 1199011198673119860


3119860

lowast=

1198760(1199013119860

lowast) = 2119887119892(119886 minus 119887ℎ)


119860(119901) = 119886 minus 119887119901


1198713119860lt 119901 lt

1199011198673119860


119864 [119902 (119901)] = 119911119860(119901) int

119906

1198760(119901)119911119860(119901)

119909119891 (119909) 119889119909

= Γ (119901)

=(119886 minus 119887119901) 119906

2

2 (119906 minus 119897)minus

1198922

2 (119906 minus 119897) (119901 minus ℎ)2

(119886 minus 119887119901)

(A20)


119889Γ (119901)

119889119901=

[1198922(2119886 + 119887ℎ minus 3119887119901) minus 119887119906

2(119901 minus ℎ)

3

(119886 minus 119887119901)2

]

2 (119906 minus 119897) (119901 minus ℎ)3

(119886 minus 119887119901)2

(A21)

1198892Γ (119901)

1198891199012

=minus1198922

3 (119906 minus 119897) (119901 minus ℎ)4

(119886 minus 119887119901)3

times [ 181198872(119901 minus

(2119886 + 119887ℎ)

3119887)

2

+ (119886 minus 119887ℎ)2]


(A22)


1199014119860=arg119901

1198922(2119886 + 119887ℎ minus 3119887119901) minus 119887119906

2(119901 minus ℎ)

3

(119886 minus 119887119901)2

= 0

(A23)

If 1199011198713119860

lt 1199014119860lt 1199011198673119860

then 119901 = 1199014119860


lowast= 1198760(1199014) = 119892(119901

4minus ℎ)


1198673119860

1198760(119901)119911119860(119901) = lim

119901rarr1199011198713119860

1198760(119901)119911119860(119901) = 119906


1198673119860

120578(119901) = lim119901rarr119901

1198713119860

120578(119901) = 0


1198673119860

119864[119902(119901)] =

lim119901rarr119901

1198713119860

119864[119902(119901)] = 0 If 1199014119860

le 1199011198713119860

or 1199014119860

ge 1199011198673119860


lt 119901 lt 1199011198673119860



this case


119887119886


1119872

lowast(119876) = 119901

0(119876) = ℎ + 119892119876



1119872(119876)119889119876 =

[ℎ + 119892119876]119887minus1ℎ + 119892(1 minus 119887)119876119886


119864 [1205871119872(119876)]

= int

119906

1198701119872(119876)

([1199011119872

lowast

(119876) minus ℎ] 119909119911119872[1199011119872

lowast

(119876)] minus 119892) 119891 (119909) 119889119909

=119892 [119906 minus 119870

1119872(119876)]

(119906 minus 119897)[119886

119876(119892

119876+ ℎ)

minus119887

(119906 + 119870

1119872(119876)

2) minus 1]

(A24)



119889119864 [1205871119872(119876)]

119889119876

= minus119886119892119876minus3[ℎ +

119892

119876]

minus119887minus1

[1199062minus 1198701119898

2

(119876)]ℎ119876 minus (119887 minus 1) 119892

2 (119906 minus 119897)

(A25)


1119872(119876)] is


1119872(0) = 0 (as

1198701119872(119876) = 119876

1119887ℎ + 119892119876


119864[1205871119872(0)] = +infin


1)119892ℎ 119889119864[1205871119872(119876)]119889119876 = 0 for 119876 = (119887 minus 1)119892ℎ and





1119872(119876)119889119876 gt



119887119886ℎ


1119872(119876) gt 119906





lowast= 1199011119872

lowast(1198761119872

lowast) = 119887ℎ(119887 minus 1)


119872(119901) = 119886119901






1198703119872(119901) =

119892119901119887

119886 (119901 minus ℎ)lt 119906 (A26)

(b) and (c)We have 1198891198703119872(119901)119889119901 = (119892119901

119887119886(119901minusℎ)

2)[(119887minus1)119901minus


is given by

Ψ3119872(119901) = 1 minus 119865[

1198760(119901)

119911119872(119901)

] (A27)

If 119897 lt 1198703119872(119901) lt 119906

Ψ3119872(119901) = minus

119892

119886 (119906 minus 119897)(

119901119887

119901 minus ℎ) + (1 +

119897

119906 minus 119897) (A28)


119889Ψ3119872(119901)

119889119901= minus

119892119901119887minus1


If 119897 ge 1198703119872(119901) Ψ

3119872(119901)= 1 and if 119906 le 119870

3119872(119901) Ψ

3119872(119901) = 0

(note that (1198763119872




unique1199013119872119897

1015840 and unique1199013119872119906

1015840 such that (i)1199013119872119897

1015840gt 1199013119872119906

1015840gt

ℎ (ii) 1198703119872(1199013119872119897

1015840) = 119897 and (iii) 119870

3119872(1199013119872119906

1015840) = 119906 Observed


119897 lt 1198703119872(119901) lt 119906 Therefore Ψ


for all 1199013119872119906

1015840lt 119901 lt 119901

3119872119897

1015840 and Ψ3119872(119897) = 1 for all 119901 ge 119901

3119872119897

1015840


3119872119906

1015840)For 119887 gt 1 119889119870


1198891198703119872(119901)119889119901 = 0 for 119901 = ℎ119887(119887minus1) and 119889119870

3119872(119901)119889119901 gt 0 for



= ℎ119887(119887 minus 1) As 1198703119872(1199013119872) =



0(119901) 119901)


119887lt 119906 as


3119872119906

10158401015840 and 1199013119872119906

101584010158401015840 such that (i)ℎ lt 119901

3119872119906

10158401015840lt 1199013119872119906

101584010158401015840 (ii) 1198703119872(1199013119872119906

10158401015840) = 119870

3119872(1199013119872119906

101584010158401015840) = 119906

and (iii) 1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 If(119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))

119887gt 119897 then 119897 lt 119870

3119872(119901) lt 119906 for

all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119906


10158401015840 and1199013119872119897

101584010158401015840 such that (i) 1199013119872119906

10158401015840lt 1199013119872119897

10158401015840le 1199013119872119897

101584010158401015840lt 1199013119872119906

101584010158401015840(ii) 119870

3119872(1199013119872119897

10158401015840) = 119870

3119872(1199013119872119897

101584010158401015840) = 119897 (iii) 119897 lt 119870

3119872(119901) lt 119906

for all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 or 1199013119872119897

101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 and(iv)1198703119872(119901) le 119897 for all 119901

3119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840Next for 119887 gt 1 and 119897 lt 119870

3119872(119901) lt 119906 119889Ψ

3119872(119901)119889119901 gt 0 for







1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897


101584010158401015840lt

119901 lt 1199013119872119906

101584010158401015840 and Ψ3119872(119901) = 1 for all 119901

3119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840


119864 [1199024119872(119901)] = Γ

1(119901) = int

119906

1198703119872(119901)

119909119891 (119909) 119889119909

=1199062

2 (119906 minus 119897)minus

11989221199012119887

21198862(119906 minus 119897) (119901 minus ℎ)

2

(A30)

for 119897 lt 1198703119872(119901) lt 119906 where119870

3119872(119901) is given by (23)

119864 [1199024119872(119901)] =

119906 minus 119897

2 for 119870

3119872(119901) le 119897

119864 [1199024119872(119901)] = 0 for 119870

3119872(119901) ge 119906

(A31)


below respectively

119889Γ1(119901)

119889119901= minus

11989221199012119887minus1

1198862(119906 minus 119897) (119901 minus ℎ)

3[(119887 minus 1) 119901 minus 119887ℎ] (A32)






3119872(119901) lt

119906 implies 1199013119872119906

1015840lt 119901 lt 119901

3119872119897

1015840 and 1198703119872(119901) le 119897 implies 119901 ge

1199013119872119897

1015840 where 1198703119872(1199013119872119897

1015840) = 119897 and 119870

3119872(1199013119872119906

1015840) = 119906 (see the


119864[1199024119872(1199013119872119897

1015840)] = (119906 minus 119897)2 for all 119901

3119872119906

1015840lt 119901 lt 119901

3119872119897


3119872119897

1015840For 119887 gt 1 and 119897 lt 119870

3119872(119901) lt 119906 119889Γ

1(119901)119889119901 gt 0 for all





(119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))119887lt 119906


1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906


119887lt 119906

119897 lt 1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906


1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 or 1199013119872119897

101584010158401015840lt

119901 lt 1199013119872119906

101584010158401015840 and 1198703119872(119901) le 119897 for all 119901

3119872119897

10158401015840lt 119901 lt

1199013119872119897





for all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897


101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 and 119864[1199024119872(119901)] = (119906 minus 119897)2 for all

1199013119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840



Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













2

1199064119887 2119887119892(119886 minus 119887ℎ) (119886 + 119887ℎ)2119887


2

1199064119887 and 1199011198714119860

lt 1199014119860lt 1199011198674119860

119892(1199014119860minus ℎ) Given by 119901

4119860









1119860

lowast= 2119892119887(119886 minus 119887ℎ) gt


1119860

lowast= (119886 + 119887ℎ)2119887 gt ℎ



1198673119860= (119886 + 119887ℎ)2119887 + radicΔ2119906119887




119887ℎ)24119887



1198763119860

lowast= 2119887119892(119886 minus 119887ℎ)


24119887



4119860= arg1199011198922(2119886+119887ℎminus3119887119901)minus119887119906

2(119901minusℎ)

3(119886minus119887119901)

2=

0 If 1199011198713119860

lt 1199014119860lt 1199011198673119860



lowast= 119892(119901

4119860minus ℎ)









1198714119860lt 1199014119860

lt

1199011198674119860



119911119872(119901) = 119886119901





1119872(119876)] is




minus119887








rium 119901 are given by 1198761119872


1199011119872


Let Δ3119860

= 119906[119906(119886 minus 119887ℎ)2minus 4119887119892] 119901

1198713119860= (119886 + 119887ℎ)2119887 minus

radicΔ31198602119906119887 and 119901

1198673119860= (119886 + 119887ℎ)2119887 +radicΔ

31198602119906119887 Moreover

let

1198703119872(119901) =

119892119901119887

119886 (119901 minus ℎ)lt 119906 (23)

and denote 1199013119872119897

1015840 1199013119872119906

1015840 1199013119872119906

10158401015840 1199013119872119906

101584010158401015840 1199013119872119897

10158401015840 1199013119872119897

1015840101584010158401199013119872119897

1015840 and 1199013119872119906

1015840 that satisfy (i) 1199013119872119897

1015840gt 1199013119872119906

1015840gt ℎ (ii)

1198703119872(1199013119872119897

1015840) = 119897 and (iii) 119870

3119872(1199013119872119906

1015840) = 119906 and 119901

3119872119906

101584010158401199013119872119906

101584010158401015840 1199013119872119897

10158401015840 and 1199013119872119897

101584010158401015840 satisfy (i) ℎ lt 1199013119872119906

10158401015840lt 1199013119872119897

10158401015840le

1199013119872119897

101584010158401015840lt 1199013119872119906

101584010158401015840 (ii) 1198703119872(1199013119872119906

10158401015840) = 119870

3119872(1199013119872119906

101584010158401015840) = 119906 (iii)

1198703119872(1199013119872119897

10158401015840) = 119870

3119872(1199013119872119897

101584010158401015840) = 119897 (iv) 119897 lt 119870

3119872(119901) lt 119906 for

all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 or 1199013119872119897

101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 and (v)1198703119872(119901) le 119897 for all 119901

3119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840




3119872119906

1015840lt 119901 lt 119901

3119872119897

1015840 and Ψ(119897)=1 for all 119901 ge 119901

3119872119897

1015840(c) for 119887 gt 1




3119872119906

10158401015840lt 119901 lt

1199013119872119897


101584010158401015840lt

119901 lt 1199013119872119906

101584010158401015840 andΨ3119872(119901) = 1 for all 119901

3119872119897

10158401015840lt 119901 lt

1199013119872119897

101584010158401015840







1015840lt119901lt119901

3119872119897

1015840 and119864[119902(119901 119876

0(119901))] = (119906 minus 119897)2 for all 119901 ge 119901

3119872119897

1015840(b) for 119887 gt 1


lt 119906then 119864[119902(119901 119876



119887lt 119897 then


1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897


3119872119897

101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 andΨ3119872(119901) = 1 for

all 1199013119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840





With 119911(119901) = 119911119872(119901) = 119886119901


119860(119901) = 119886 minus 119887119901







6 Discussion



Appendix

Mathematical Proofs




119889Ψ (119901)

119889119901= 119891[

119876

119911 (119901)] [

1198761199111015840(119901)

1199112(119901)

] le 0 as 1199111015840 (119901) lt 0

119891 (119909) ge 0 forall119909 ge 0

(A1)


0(119876) by (C1)


lowast(119876) = 119901



119864 [120587 (119876)] =120572int

infin

119876119911[1199012

lowast(119876)]

[(119892

119876)119911 (119901

2

lowast

(119876)) 119909 minus 119892]119891 (119909) 119889119909

= 120572119892[119911 (ℎ + 119892119876)

119876int

infin

119876119911(ℎ+119892119876)

119909119891 (119909) 119889119909

+ 119865(119876

119911 (ℎ + 119892119876)) minus 1]

(A2)

119889119864[120587 (119876)]

119889119876= minus(

120572119892

1198763)[119876119911(ℎ +

119892

119876) + 119892119911

1015840(ℎ +

119892

119876)]

times int

infin

119876119911(ℎ+119892119876)

119909119891 (119909) 119889119909

(A3)


119876119911(ℎ+119892119876)119909119891(119909)119889119909 gt (119876119911(ℎ+


119876=1198762


lowast119911(ℎ + 119892119876

2

lowast) + 119892119911

1015840(ℎ +

1198921198762

lowast) = 0



infin

119876119911(119901)119909119891(119909)119889119909 and

we have 119889119864[119902(119901)]119889119901 = 1199111015840(119901)[intinfin

119876119911(119901)119909119891(119909)119889119909 +

1198762119891[119876119911(119901)]119911

2(119901)] le 0 as 1199111015840(119901) lt 0 and 119891(119909) ge





119864 [120587 (119876)] = 120572int

infin

119876119911(119901)

[(119901 minus ℎ) 119911 (119901) 119909 minus 119892] 119891 (119909) 119889119909

= 120572((119901 minus ℎ) 119911 (119901)int

infin

119876119911(119901)

119909119891 (119909) 119889119909

+119892119865[119876

119911 (119901)] + 119892)

(A4)



119889119864 [120587 (119876)]

119889119876= minus

120572119891 [119876119911 (119901)] [(119901 minus ℎ)119876 minus 119892]

119911 (119901) (A5)

1198892119864 [120587 (119876)]

1198891198762

= minus120572((119901 minus ℎ) 119911 (119901) 119891[119876

119911 (119901)]

+ [(119901 minus ℎ)119876 minus 119892]1198911015840[

119876

119911 (119901)]) times (119911

2(119901))

minus1

(A6)


(119901 minus ℎ) 119911 (119901) 119891 [119876

119911 (119901)] + [(119901 minus ℎ)119876 minus 119892]119891

1015840[

119876

119911 (119901)]

ge (119901 minus ℎ) 119911 (119901) 119891 [119876

119911 (119901)] gt 0

(A7)




3


is given by

Ψ (119901) = 1 minus 119865[1198763

lowast(119901)

119911 (119901)] = 1 minus 119865[

119892

(119901 minus ℎ) 119911 (119901)] (A8)


119889Ψ (119901)

119889119901= [

119892

(119901 minus ℎ)2

1199112(119901)

] [119911 (119901) + (119901 minus ℎ) 1199111015840(119901)]

times 119891[119892

(119901 minus ℎ) 119911 (119901)]

(A9)



119911(1199013

lowast) + (119901

3


1015840(1199013


(A9)



4




is in the form

119889119864 [119902 (119901)]

119889119901=

120577 (119901)

[(119901 minus ℎ)3

1199112(119901)]

(A10)


4


4






1119860(119876) where 119870

1119860(119876) =

119876119911119860(1199011119860


1119860(119876) lt 119906


1198762minus (119886 minus 119887ℎ) 119906119876 + 119887119892119906

gt 1198762minus (119886 minus 119887ℎ) 119906119876 +

(119886 minus 119887ℎ)21199062

4

= [119876 minus(119886 minus 119887ℎ) 119906

2]

2

ge 0

(A11)


given by

1205871119860(119876)

=

[1199011119860

lowast(119876) minus ℎ] 119909119911

119860[1199011119860


1119860(119876)

0 ow(A12)

where 1199011119860

lowast(119876) = 119901



119864 [1205871119860(119876)]

= int

119906

1198701119860(119876)

([1199011119860

lowast

(119876) minus ℎ] 119909119911119860[1199011119860

lowast

(119876)] minus 119892) 119891 (119909) 119889119909

=119892 [119906 minus 119870

1119860(119876)]

(119906 minus 119897)([119886 minus 119887ℎ

119876minus119887119892

1198762] [

119906 + 1198701119860(119876)

2] minus 1)

(A13)



119889119864 [1205871119860(119876)]

119889119876=

119892 [1199062minus 1198701119860

2(119876)] [minus (119886 minus 119887ℎ)119876 + 2119887119892]

2 (119906 minus 119897) 1198763

(A14)

1198892119864 [1205871119860(119876)]

1198891198762

=

119892 [1199062minus 1198701119860

2(119876)] [(119886 minus 119887ℎ)119876 minus 3119887119892]

(119906 minus 119897) 1198764

(A15)

From (A15) for 119876 gt 0 1198892119864[1205871119860(119876)]119889119876


3119887119892(119886 minus 119887ℎ) (as 1199062 gt 1198701119860

2(119876)) Thus 119864[120587

1119860(119876)] is strictly





1119860(119876)]119889119876 = 0) Therefore 119864[120587

1119860(119876)]


lowast= 2119887119892(119886 minus 119887ℎ) is


lowast= 1199011119860

lowast(1198761119860

lowast) =

(119886 + 119887ℎ)2119887


119860(119901) = 119886 minus 119887119901





120578 (119901) = 1199061198871199012minus 119906 (119886 + 119887ℎ) 119901 + 119906ℎ119886 + 119892 lt 0 (A16)


lt 119901 lt

1199011198673119860


Ψ3(119901) = 1 minus 119865[

1198763119860

lowast(119901)

119911119860(119901)

]

= minus119892


119906

(119906 minus 119897)]

(A17)



119889Ψ (119901)

119889119901=

119892

(119906 minus 119897) (119901 minus ℎ)2

(119886 minus 119887119901)2(119886 + ℎ119887 minus 2119887119901) (A18)

1198892Ψ (119901)

1198891199012

= minus2119892

(119906 minus 119897) (119901 minus ℎ)3

(119886 minus 119887119901)3


] lt 0

(A19)

for all 1199011198713119860

lt 119901 lt 1199011198673119860


for 1199011198713119860

lt 119901 lt 1199011198673119860


3119860

lowast= (119886 + ℎ119887)2119887


lt 1199013119860

lowastlt 1199011198673119860


3119860

lowast=

1198760(1199013119860

lowast) = 2119887119892(119886 minus 119887ℎ)


119860(119901) = 119886 minus 119887119901


1198713119860lt 119901 lt

1199011198673119860


119864 [119902 (119901)] = 119911119860(119901) int

119906

1198760(119901)119911119860(119901)

119909119891 (119909) 119889119909

= Γ (119901)

=(119886 minus 119887119901) 119906

2

2 (119906 minus 119897)minus

1198922

2 (119906 minus 119897) (119901 minus ℎ)2

(119886 minus 119887119901)

(A20)


119889Γ (119901)

119889119901=

[1198922(2119886 + 119887ℎ minus 3119887119901) minus 119887119906

2(119901 minus ℎ)

3

(119886 minus 119887119901)2

]

2 (119906 minus 119897) (119901 minus ℎ)3

(119886 minus 119887119901)2

(A21)

1198892Γ (119901)

1198891199012

=minus1198922

3 (119906 minus 119897) (119901 minus ℎ)4

(119886 minus 119887119901)3

times [ 181198872(119901 minus

(2119886 + 119887ℎ)

3119887)

2

+ (119886 minus 119887ℎ)2]


(A22)


1199014119860=arg119901

1198922(2119886 + 119887ℎ minus 3119887119901) minus 119887119906

2(119901 minus ℎ)

3

(119886 minus 119887119901)2

= 0

(A23)

If 1199011198713119860

lt 1199014119860lt 1199011198673119860

then 119901 = 1199014119860


lowast= 1198760(1199014) = 119892(119901

4minus ℎ)


1198673119860

1198760(119901)119911119860(119901) = lim

119901rarr1199011198713119860

1198760(119901)119911119860(119901) = 119906


1198673119860

120578(119901) = lim119901rarr119901

1198713119860

120578(119901) = 0


1198673119860

119864[119902(119901)] =

lim119901rarr119901

1198713119860

119864[119902(119901)] = 0 If 1199014119860

le 1199011198713119860

or 1199014119860

ge 1199011198673119860


lt 119901 lt 1199011198673119860



this case


119887119886


1119872

lowast(119876) = 119901

0(119876) = ℎ + 119892119876



1119872(119876)119889119876 =

[ℎ + 119892119876]119887minus1ℎ + 119892(1 minus 119887)119876119886


119864 [1205871119872(119876)]

= int

119906

1198701119872(119876)

([1199011119872

lowast

(119876) minus ℎ] 119909119911119872[1199011119872

lowast

(119876)] minus 119892) 119891 (119909) 119889119909

=119892 [119906 minus 119870

1119872(119876)]

(119906 minus 119897)[119886

119876(119892

119876+ ℎ)

minus119887

(119906 + 119870

1119872(119876)

2) minus 1]

(A24)



119889119864 [1205871119872(119876)]

119889119876

= minus119886119892119876minus3[ℎ +

119892

119876]

minus119887minus1

[1199062minus 1198701119898

2

(119876)]ℎ119876 minus (119887 minus 1) 119892

2 (119906 minus 119897)

(A25)


1119872(119876)] is


1119872(0) = 0 (as

1198701119872(119876) = 119876

1119887ℎ + 119892119876


119864[1205871119872(0)] = +infin


1)119892ℎ 119889119864[1205871119872(119876)]119889119876 = 0 for 119876 = (119887 minus 1)119892ℎ and





1119872(119876)119889119876 gt



119887119886ℎ


1119872(119876) gt 119906





lowast= 1199011119872

lowast(1198761119872

lowast) = 119887ℎ(119887 minus 1)


119872(119901) = 119886119901






1198703119872(119901) =

119892119901119887

119886 (119901 minus ℎ)lt 119906 (A26)

(b) and (c)We have 1198891198703119872(119901)119889119901 = (119892119901

119887119886(119901minusℎ)

2)[(119887minus1)119901minus


is given by

Ψ3119872(119901) = 1 minus 119865[

1198760(119901)

119911119872(119901)

] (A27)

If 119897 lt 1198703119872(119901) lt 119906

Ψ3119872(119901) = minus

119892

119886 (119906 minus 119897)(

119901119887

119901 minus ℎ) + (1 +

119897

119906 minus 119897) (A28)


119889Ψ3119872(119901)

119889119901= minus

119892119901119887minus1


If 119897 ge 1198703119872(119901) Ψ

3119872(119901)= 1 and if 119906 le 119870

3119872(119901) Ψ

3119872(119901) = 0

(note that (1198763119872




unique1199013119872119897

1015840 and unique1199013119872119906

1015840 such that (i)1199013119872119897

1015840gt 1199013119872119906

1015840gt

ℎ (ii) 1198703119872(1199013119872119897

1015840) = 119897 and (iii) 119870

3119872(1199013119872119906

1015840) = 119906 Observed


119897 lt 1198703119872(119901) lt 119906 Therefore Ψ


for all 1199013119872119906

1015840lt 119901 lt 119901

3119872119897

1015840 and Ψ3119872(119897) = 1 for all 119901 ge 119901

3119872119897

1015840


3119872119906

1015840)For 119887 gt 1 119889119870


1198891198703119872(119901)119889119901 = 0 for 119901 = ℎ119887(119887minus1) and 119889119870

3119872(119901)119889119901 gt 0 for



= ℎ119887(119887 minus 1) As 1198703119872(1199013119872) =



0(119901) 119901)


119887lt 119906 as


3119872119906

10158401015840 and 1199013119872119906

101584010158401015840 such that (i)ℎ lt 119901

3119872119906

10158401015840lt 1199013119872119906

101584010158401015840 (ii) 1198703119872(1199013119872119906

10158401015840) = 119870

3119872(1199013119872119906

101584010158401015840) = 119906

and (iii) 1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 If(119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))

119887gt 119897 then 119897 lt 119870

3119872(119901) lt 119906 for

all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119906


10158401015840 and1199013119872119897

101584010158401015840 such that (i) 1199013119872119906

10158401015840lt 1199013119872119897

10158401015840le 1199013119872119897

101584010158401015840lt 1199013119872119906

101584010158401015840(ii) 119870

3119872(1199013119872119897

10158401015840) = 119870

3119872(1199013119872119897

101584010158401015840) = 119897 (iii) 119897 lt 119870

3119872(119901) lt 119906

for all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 or 1199013119872119897

101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 and(iv)1198703119872(119901) le 119897 for all 119901

3119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840Next for 119887 gt 1 and 119897 lt 119870

3119872(119901) lt 119906 119889Ψ

3119872(119901)119889119901 gt 0 for







1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897


101584010158401015840lt

119901 lt 1199013119872119906

101584010158401015840 and Ψ3119872(119901) = 1 for all 119901

3119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840


119864 [1199024119872(119901)] = Γ

1(119901) = int

119906

1198703119872(119901)

119909119891 (119909) 119889119909

=1199062

2 (119906 minus 119897)minus

11989221199012119887

21198862(119906 minus 119897) (119901 minus ℎ)

2

(A30)

for 119897 lt 1198703119872(119901) lt 119906 where119870

3119872(119901) is given by (23)

119864 [1199024119872(119901)] =

119906 minus 119897

2 for 119870

3119872(119901) le 119897

119864 [1199024119872(119901)] = 0 for 119870

3119872(119901) ge 119906

(A31)


below respectively

119889Γ1(119901)

119889119901= minus

11989221199012119887minus1

1198862(119906 minus 119897) (119901 minus ℎ)

3[(119887 minus 1) 119901 minus 119887ℎ] (A32)






3119872(119901) lt

119906 implies 1199013119872119906

1015840lt 119901 lt 119901

3119872119897

1015840 and 1198703119872(119901) le 119897 implies 119901 ge

1199013119872119897

1015840 where 1198703119872(1199013119872119897

1015840) = 119897 and 119870

3119872(1199013119872119906

1015840) = 119906 (see the


119864[1199024119872(1199013119872119897

1015840)] = (119906 minus 119897)2 for all 119901

3119872119906

1015840lt 119901 lt 119901

3119872119897


3119872119897

1015840For 119887 gt 1 and 119897 lt 119870

3119872(119901) lt 119906 119889Γ

1(119901)119889119901 gt 0 for all





(119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))119887lt 119906


1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906


119887lt 119906

119897 lt 1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906


1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 or 1199013119872119897

101584010158401015840lt

119901 lt 1199013119872119906

101584010158401015840 and 1198703119872(119901) le 119897 for all 119901

3119872119897

10158401015840lt 119901 lt

1199013119872119897





for all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897


101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 and 119864[1199024119872(119901)] = (119906 minus 119897)2 for all

1199013119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840



Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











minus119887








rium 119901 are given by 1198761119872


1199011119872


Let Δ3119860

= 119906[119906(119886 minus 119887ℎ)2minus 4119887119892] 119901

1198713119860= (119886 + 119887ℎ)2119887 minus

radicΔ31198602119906119887 and 119901

1198673119860= (119886 + 119887ℎ)2119887 +radicΔ

31198602119906119887 Moreover

let

1198703119872(119901) =

119892119901119887

119886 (119901 minus ℎ)lt 119906 (23)

and denote 1199013119872119897

1015840 1199013119872119906

1015840 1199013119872119906

10158401015840 1199013119872119906

101584010158401015840 1199013119872119897

10158401015840 1199013119872119897

1015840101584010158401199013119872119897

1015840 and 1199013119872119906

1015840 that satisfy (i) 1199013119872119897

1015840gt 1199013119872119906

1015840gt ℎ (ii)

1198703119872(1199013119872119897

1015840) = 119897 and (iii) 119870

3119872(1199013119872119906

1015840) = 119906 and 119901

3119872119906

101584010158401199013119872119906

101584010158401015840 1199013119872119897

10158401015840 and 1199013119872119897

101584010158401015840 satisfy (i) ℎ lt 1199013119872119906

10158401015840lt 1199013119872119897

10158401015840le

1199013119872119897

101584010158401015840lt 1199013119872119906

101584010158401015840 (ii) 1198703119872(1199013119872119906

10158401015840) = 119870

3119872(1199013119872119906

101584010158401015840) = 119906 (iii)

1198703119872(1199013119872119897

10158401015840) = 119870

3119872(1199013119872119897

101584010158401015840) = 119897 (iv) 119897 lt 119870

3119872(119901) lt 119906 for

all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 or 1199013119872119897

101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 and (v)1198703119872(119901) le 119897 for all 119901

3119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840




3119872119906

1015840lt 119901 lt 119901

3119872119897

1015840 and Ψ(119897)=1 for all 119901 ge 119901

3119872119897

1015840(c) for 119887 gt 1




3119872119906

10158401015840lt 119901 lt

1199013119872119897


101584010158401015840lt

119901 lt 1199013119872119906

101584010158401015840 andΨ3119872(119901) = 1 for all 119901

3119872119897

10158401015840lt 119901 lt

1199013119872119897

101584010158401015840







1015840lt119901lt119901

3119872119897

1015840 and119864[119902(119901 119876

0(119901))] = (119906 minus 119897)2 for all 119901 ge 119901

3119872119897

1015840(b) for 119887 gt 1


lt 119906then 119864[119902(119901 119876



119887lt 119897 then


1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897


3119872119897

101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 andΨ3119872(119901) = 1 for

all 1199013119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840





With 119911(119901) = 119911119872(119901) = 119886119901


119860(119901) = 119886 minus 119887119901







6 Discussion



Appendix

Mathematical Proofs




119889Ψ (119901)

119889119901= 119891[

119876

119911 (119901)] [

1198761199111015840(119901)

1199112(119901)

] le 0 as 1199111015840 (119901) lt 0

119891 (119909) ge 0 forall119909 ge 0

(A1)


0(119876) by (C1)


lowast(119876) = 119901



119864 [120587 (119876)] =120572int

infin

119876119911[1199012

lowast(119876)]

[(119892

119876)119911 (119901

2

lowast

(119876)) 119909 minus 119892]119891 (119909) 119889119909

= 120572119892[119911 (ℎ + 119892119876)

119876int

infin

119876119911(ℎ+119892119876)

119909119891 (119909) 119889119909

+ 119865(119876

119911 (ℎ + 119892119876)) minus 1]

(A2)

119889119864[120587 (119876)]

119889119876= minus(

120572119892

1198763)[119876119911(ℎ +

119892

119876) + 119892119911

1015840(ℎ +

119892

119876)]

times int

infin

119876119911(ℎ+119892119876)

119909119891 (119909) 119889119909

(A3)


119876119911(ℎ+119892119876)119909119891(119909)119889119909 gt (119876119911(ℎ+


119876=1198762


lowast119911(ℎ + 119892119876

2

lowast) + 119892119911

1015840(ℎ +

1198921198762

lowast) = 0



infin

119876119911(119901)119909119891(119909)119889119909 and

we have 119889119864[119902(119901)]119889119901 = 1199111015840(119901)[intinfin

119876119911(119901)119909119891(119909)119889119909 +

1198762119891[119876119911(119901)]119911

2(119901)] le 0 as 1199111015840(119901) lt 0 and 119891(119909) ge





119864 [120587 (119876)] = 120572int

infin

119876119911(119901)

[(119901 minus ℎ) 119911 (119901) 119909 minus 119892] 119891 (119909) 119889119909

= 120572((119901 minus ℎ) 119911 (119901)int

infin

119876119911(119901)

119909119891 (119909) 119889119909

+119892119865[119876

119911 (119901)] + 119892)

(A4)



119889119864 [120587 (119876)]

119889119876= minus

120572119891 [119876119911 (119901)] [(119901 minus ℎ)119876 minus 119892]

119911 (119901) (A5)

1198892119864 [120587 (119876)]

1198891198762

= minus120572((119901 minus ℎ) 119911 (119901) 119891[119876

119911 (119901)]

+ [(119901 minus ℎ)119876 minus 119892]1198911015840[

119876

119911 (119901)]) times (119911

2(119901))

minus1

(A6)


(119901 minus ℎ) 119911 (119901) 119891 [119876

119911 (119901)] + [(119901 minus ℎ)119876 minus 119892]119891

1015840[

119876

119911 (119901)]

ge (119901 minus ℎ) 119911 (119901) 119891 [119876

119911 (119901)] gt 0

(A7)




3


is given by

Ψ (119901) = 1 minus 119865[1198763

lowast(119901)

119911 (119901)] = 1 minus 119865[

119892

(119901 minus ℎ) 119911 (119901)] (A8)


119889Ψ (119901)

119889119901= [

119892

(119901 minus ℎ)2

1199112(119901)

] [119911 (119901) + (119901 minus ℎ) 1199111015840(119901)]

times 119891[119892

(119901 minus ℎ) 119911 (119901)]

(A9)



119911(1199013

lowast) + (119901

3


1015840(1199013


(A9)



4




is in the form

119889119864 [119902 (119901)]

119889119901=

120577 (119901)

[(119901 minus ℎ)3

1199112(119901)]

(A10)


4


4






1119860(119876) where 119870

1119860(119876) =

119876119911119860(1199011119860


1119860(119876) lt 119906


1198762minus (119886 minus 119887ℎ) 119906119876 + 119887119892119906

gt 1198762minus (119886 minus 119887ℎ) 119906119876 +

(119886 minus 119887ℎ)21199062

4

= [119876 minus(119886 minus 119887ℎ) 119906

2]

2

ge 0

(A11)


given by

1205871119860(119876)

=

[1199011119860

lowast(119876) minus ℎ] 119909119911

119860[1199011119860


1119860(119876)

0 ow(A12)

where 1199011119860

lowast(119876) = 119901



119864 [1205871119860(119876)]

= int

119906

1198701119860(119876)

([1199011119860

lowast

(119876) minus ℎ] 119909119911119860[1199011119860

lowast

(119876)] minus 119892) 119891 (119909) 119889119909

=119892 [119906 minus 119870

1119860(119876)]

(119906 minus 119897)([119886 minus 119887ℎ

119876minus119887119892

1198762] [

119906 + 1198701119860(119876)

2] minus 1)

(A13)



119889119864 [1205871119860(119876)]

119889119876=

119892 [1199062minus 1198701119860

2(119876)] [minus (119886 minus 119887ℎ)119876 + 2119887119892]

2 (119906 minus 119897) 1198763

(A14)

1198892119864 [1205871119860(119876)]

1198891198762

=

119892 [1199062minus 1198701119860

2(119876)] [(119886 minus 119887ℎ)119876 minus 3119887119892]

(119906 minus 119897) 1198764

(A15)

From (A15) for 119876 gt 0 1198892119864[1205871119860(119876)]119889119876


3119887119892(119886 minus 119887ℎ) (as 1199062 gt 1198701119860

2(119876)) Thus 119864[120587

1119860(119876)] is strictly





1119860(119876)]119889119876 = 0) Therefore 119864[120587

1119860(119876)]


lowast= 2119887119892(119886 minus 119887ℎ) is


lowast= 1199011119860

lowast(1198761119860

lowast) =

(119886 + 119887ℎ)2119887


119860(119901) = 119886 minus 119887119901





120578 (119901) = 1199061198871199012minus 119906 (119886 + 119887ℎ) 119901 + 119906ℎ119886 + 119892 lt 0 (A16)


lt 119901 lt

1199011198673119860


Ψ3(119901) = 1 minus 119865[

1198763119860

lowast(119901)

119911119860(119901)

]

= minus119892


119906

(119906 minus 119897)]

(A17)



119889Ψ (119901)

119889119901=

119892

(119906 minus 119897) (119901 minus ℎ)2

(119886 minus 119887119901)2(119886 + ℎ119887 minus 2119887119901) (A18)

1198892Ψ (119901)

1198891199012

= minus2119892

(119906 minus 119897) (119901 minus ℎ)3

(119886 minus 119887119901)3


] lt 0

(A19)

for all 1199011198713119860

lt 119901 lt 1199011198673119860


for 1199011198713119860

lt 119901 lt 1199011198673119860


3119860

lowast= (119886 + ℎ119887)2119887


lt 1199013119860

lowastlt 1199011198673119860


3119860

lowast=

1198760(1199013119860

lowast) = 2119887119892(119886 minus 119887ℎ)


119860(119901) = 119886 minus 119887119901


1198713119860lt 119901 lt

1199011198673119860


119864 [119902 (119901)] = 119911119860(119901) int

119906

1198760(119901)119911119860(119901)

119909119891 (119909) 119889119909

= Γ (119901)

=(119886 minus 119887119901) 119906

2

2 (119906 minus 119897)minus

1198922

2 (119906 minus 119897) (119901 minus ℎ)2

(119886 minus 119887119901)

(A20)


119889Γ (119901)

119889119901=

[1198922(2119886 + 119887ℎ minus 3119887119901) minus 119887119906

2(119901 minus ℎ)

3

(119886 minus 119887119901)2

]

2 (119906 minus 119897) (119901 minus ℎ)3

(119886 minus 119887119901)2

(A21)

1198892Γ (119901)

1198891199012

=minus1198922

3 (119906 minus 119897) (119901 minus ℎ)4

(119886 minus 119887119901)3

times [ 181198872(119901 minus

(2119886 + 119887ℎ)

3119887)

2

+ (119886 minus 119887ℎ)2]


(A22)


1199014119860=arg119901

1198922(2119886 + 119887ℎ minus 3119887119901) minus 119887119906

2(119901 minus ℎ)

3

(119886 minus 119887119901)2

= 0

(A23)

If 1199011198713119860

lt 1199014119860lt 1199011198673119860

then 119901 = 1199014119860


lowast= 1198760(1199014) = 119892(119901

4minus ℎ)


1198673119860

1198760(119901)119911119860(119901) = lim

119901rarr1199011198713119860

1198760(119901)119911119860(119901) = 119906


1198673119860

120578(119901) = lim119901rarr119901

1198713119860

120578(119901) = 0


1198673119860

119864[119902(119901)] =

lim119901rarr119901

1198713119860

119864[119902(119901)] = 0 If 1199014119860

le 1199011198713119860

or 1199014119860

ge 1199011198673119860


lt 119901 lt 1199011198673119860



this case


119887119886


1119872

lowast(119876) = 119901

0(119876) = ℎ + 119892119876



1119872(119876)119889119876 =

[ℎ + 119892119876]119887minus1ℎ + 119892(1 minus 119887)119876119886


119864 [1205871119872(119876)]

= int

119906

1198701119872(119876)

([1199011119872

lowast

(119876) minus ℎ] 119909119911119872[1199011119872

lowast

(119876)] minus 119892) 119891 (119909) 119889119909

=119892 [119906 minus 119870

1119872(119876)]

(119906 minus 119897)[119886

119876(119892

119876+ ℎ)

minus119887

(119906 + 119870

1119872(119876)

2) minus 1]

(A24)



119889119864 [1205871119872(119876)]

119889119876

= minus119886119892119876minus3[ℎ +

119892

119876]

minus119887minus1

[1199062minus 1198701119898

2

(119876)]ℎ119876 minus (119887 minus 1) 119892

2 (119906 minus 119897)

(A25)


1119872(119876)] is


1119872(0) = 0 (as

1198701119872(119876) = 119876

1119887ℎ + 119892119876


119864[1205871119872(0)] = +infin


1)119892ℎ 119889119864[1205871119872(119876)]119889119876 = 0 for 119876 = (119887 minus 1)119892ℎ and





1119872(119876)119889119876 gt



119887119886ℎ


1119872(119876) gt 119906





lowast= 1199011119872

lowast(1198761119872

lowast) = 119887ℎ(119887 minus 1)


119872(119901) = 119886119901






1198703119872(119901) =

119892119901119887

119886 (119901 minus ℎ)lt 119906 (A26)

(b) and (c)We have 1198891198703119872(119901)119889119901 = (119892119901

119887119886(119901minusℎ)

2)[(119887minus1)119901minus


is given by

Ψ3119872(119901) = 1 minus 119865[

1198760(119901)

119911119872(119901)

] (A27)

If 119897 lt 1198703119872(119901) lt 119906

Ψ3119872(119901) = minus

119892

119886 (119906 minus 119897)(

119901119887

119901 minus ℎ) + (1 +

119897

119906 minus 119897) (A28)


119889Ψ3119872(119901)

119889119901= minus

119892119901119887minus1


If 119897 ge 1198703119872(119901) Ψ

3119872(119901)= 1 and if 119906 le 119870

3119872(119901) Ψ

3119872(119901) = 0

(note that (1198763119872




unique1199013119872119897

1015840 and unique1199013119872119906

1015840 such that (i)1199013119872119897

1015840gt 1199013119872119906

1015840gt

ℎ (ii) 1198703119872(1199013119872119897

1015840) = 119897 and (iii) 119870

3119872(1199013119872119906

1015840) = 119906 Observed


119897 lt 1198703119872(119901) lt 119906 Therefore Ψ


for all 1199013119872119906

1015840lt 119901 lt 119901

3119872119897

1015840 and Ψ3119872(119897) = 1 for all 119901 ge 119901

3119872119897

1015840


3119872119906

1015840)For 119887 gt 1 119889119870


1198891198703119872(119901)119889119901 = 0 for 119901 = ℎ119887(119887minus1) and 119889119870

3119872(119901)119889119901 gt 0 for



= ℎ119887(119887 minus 1) As 1198703119872(1199013119872) =



0(119901) 119901)


119887lt 119906 as


3119872119906

10158401015840 and 1199013119872119906

101584010158401015840 such that (i)ℎ lt 119901

3119872119906

10158401015840lt 1199013119872119906

101584010158401015840 (ii) 1198703119872(1199013119872119906

10158401015840) = 119870

3119872(1199013119872119906

101584010158401015840) = 119906

and (iii) 1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 If(119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))

119887gt 119897 then 119897 lt 119870

3119872(119901) lt 119906 for

all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119906


10158401015840 and1199013119872119897

101584010158401015840 such that (i) 1199013119872119906

10158401015840lt 1199013119872119897

10158401015840le 1199013119872119897

101584010158401015840lt 1199013119872119906

101584010158401015840(ii) 119870

3119872(1199013119872119897

10158401015840) = 119870

3119872(1199013119872119897

101584010158401015840) = 119897 (iii) 119897 lt 119870

3119872(119901) lt 119906

for all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 or 1199013119872119897

101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 and(iv)1198703119872(119901) le 119897 for all 119901

3119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840Next for 119887 gt 1 and 119897 lt 119870

3119872(119901) lt 119906 119889Ψ

3119872(119901)119889119901 gt 0 for







1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897


101584010158401015840lt

119901 lt 1199013119872119906

101584010158401015840 and Ψ3119872(119901) = 1 for all 119901

3119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840


119864 [1199024119872(119901)] = Γ

1(119901) = int

119906

1198703119872(119901)

119909119891 (119909) 119889119909

=1199062

2 (119906 minus 119897)minus

11989221199012119887

21198862(119906 minus 119897) (119901 minus ℎ)

2

(A30)

for 119897 lt 1198703119872(119901) lt 119906 where119870

3119872(119901) is given by (23)

119864 [1199024119872(119901)] =

119906 minus 119897

2 for 119870

3119872(119901) le 119897

119864 [1199024119872(119901)] = 0 for 119870

3119872(119901) ge 119906

(A31)


below respectively

119889Γ1(119901)

119889119901= minus

11989221199012119887minus1

1198862(119906 minus 119897) (119901 minus ℎ)

3[(119887 minus 1) 119901 minus 119887ℎ] (A32)






3119872(119901) lt

119906 implies 1199013119872119906

1015840lt 119901 lt 119901

3119872119897

1015840 and 1198703119872(119901) le 119897 implies 119901 ge

1199013119872119897

1015840 where 1198703119872(1199013119872119897

1015840) = 119897 and 119870

3119872(1199013119872119906

1015840) = 119906 (see the


119864[1199024119872(1199013119872119897

1015840)] = (119906 minus 119897)2 for all 119901

3119872119906

1015840lt 119901 lt 119901

3119872119897


3119872119897

1015840For 119887 gt 1 and 119897 lt 119870

3119872(119901) lt 119906 119889Γ

1(119901)119889119901 gt 0 for all





(119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))119887lt 119906


1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906


119887lt 119906

119897 lt 1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906


1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 or 1199013119872119897

101584010158401015840lt

119901 lt 1199013119872119906

101584010158401015840 and 1198703119872(119901) le 119897 for all 119901

3119872119897

10158401015840lt 119901 lt

1199013119872119897





for all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897


101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 and 119864[1199024119872(119901)] = (119906 minus 119897)2 for all

1199013119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840



Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











6 Discussion



Appendix

Mathematical Proofs




119889Ψ (119901)

119889119901= 119891[

119876

119911 (119901)] [

1198761199111015840(119901)

1199112(119901)

] le 0 as 1199111015840 (119901) lt 0

119891 (119909) ge 0 forall119909 ge 0

(A1)


0(119876) by (C1)


lowast(119876) = 119901



119864 [120587 (119876)] =120572int

infin

119876119911[1199012

lowast(119876)]

[(119892

119876)119911 (119901

2

lowast

(119876)) 119909 minus 119892]119891 (119909) 119889119909

= 120572119892[119911 (ℎ + 119892119876)

119876int

infin

119876119911(ℎ+119892119876)

119909119891 (119909) 119889119909

+ 119865(119876

119911 (ℎ + 119892119876)) minus 1]

(A2)

119889119864[120587 (119876)]

119889119876= minus(

120572119892

1198763)[119876119911(ℎ +

119892

119876) + 119892119911

1015840(ℎ +

119892

119876)]

times int

infin

119876119911(ℎ+119892119876)

119909119891 (119909) 119889119909

(A3)


119876119911(ℎ+119892119876)119909119891(119909)119889119909 gt (119876119911(ℎ+


119876=1198762


lowast119911(ℎ + 119892119876

2

lowast) + 119892119911

1015840(ℎ +

1198921198762

lowast) = 0



infin

119876119911(119901)119909119891(119909)119889119909 and

we have 119889119864[119902(119901)]119889119901 = 1199111015840(119901)[intinfin

119876119911(119901)119909119891(119909)119889119909 +

1198762119891[119876119911(119901)]119911

2(119901)] le 0 as 1199111015840(119901) lt 0 and 119891(119909) ge





119864 [120587 (119876)] = 120572int

infin

119876119911(119901)

[(119901 minus ℎ) 119911 (119901) 119909 minus 119892] 119891 (119909) 119889119909

= 120572((119901 minus ℎ) 119911 (119901)int

infin

119876119911(119901)

119909119891 (119909) 119889119909

+119892119865[119876

119911 (119901)] + 119892)

(A4)



119889119864 [120587 (119876)]

119889119876= minus

120572119891 [119876119911 (119901)] [(119901 minus ℎ)119876 minus 119892]

119911 (119901) (A5)

1198892119864 [120587 (119876)]

1198891198762

= minus120572((119901 minus ℎ) 119911 (119901) 119891[119876

119911 (119901)]

+ [(119901 minus ℎ)119876 minus 119892]1198911015840[

119876

119911 (119901)]) times (119911

2(119901))

minus1

(A6)


(119901 minus ℎ) 119911 (119901) 119891 [119876

119911 (119901)] + [(119901 minus ℎ)119876 minus 119892]119891

1015840[

119876

119911 (119901)]

ge (119901 minus ℎ) 119911 (119901) 119891 [119876

119911 (119901)] gt 0

(A7)




3


is given by

Ψ (119901) = 1 minus 119865[1198763

lowast(119901)

119911 (119901)] = 1 minus 119865[

119892

(119901 minus ℎ) 119911 (119901)] (A8)


119889Ψ (119901)

119889119901= [

119892

(119901 minus ℎ)2

1199112(119901)

] [119911 (119901) + (119901 minus ℎ) 1199111015840(119901)]

times 119891[119892

(119901 minus ℎ) 119911 (119901)]

(A9)



119911(1199013

lowast) + (119901

3


1015840(1199013


(A9)



4




is in the form

119889119864 [119902 (119901)]

119889119901=

120577 (119901)

[(119901 minus ℎ)3

1199112(119901)]

(A10)


4


4






1119860(119876) where 119870

1119860(119876) =

119876119911119860(1199011119860


1119860(119876) lt 119906


1198762minus (119886 minus 119887ℎ) 119906119876 + 119887119892119906

gt 1198762minus (119886 minus 119887ℎ) 119906119876 +

(119886 minus 119887ℎ)21199062

4

= [119876 minus(119886 minus 119887ℎ) 119906

2]

2

ge 0

(A11)


given by

1205871119860(119876)

=

[1199011119860

lowast(119876) minus ℎ] 119909119911

119860[1199011119860


1119860(119876)

0 ow(A12)

where 1199011119860

lowast(119876) = 119901



119864 [1205871119860(119876)]

= int

119906

1198701119860(119876)

([1199011119860

lowast

(119876) minus ℎ] 119909119911119860[1199011119860

lowast

(119876)] minus 119892) 119891 (119909) 119889119909

=119892 [119906 minus 119870

1119860(119876)]

(119906 minus 119897)([119886 minus 119887ℎ

119876minus119887119892

1198762] [

119906 + 1198701119860(119876)

2] minus 1)

(A13)



119889119864 [1205871119860(119876)]

119889119876=

119892 [1199062minus 1198701119860

2(119876)] [minus (119886 minus 119887ℎ)119876 + 2119887119892]

2 (119906 minus 119897) 1198763

(A14)

1198892119864 [1205871119860(119876)]

1198891198762

=

119892 [1199062minus 1198701119860

2(119876)] [(119886 minus 119887ℎ)119876 minus 3119887119892]

(119906 minus 119897) 1198764

(A15)

From (A15) for 119876 gt 0 1198892119864[1205871119860(119876)]119889119876


3119887119892(119886 minus 119887ℎ) (as 1199062 gt 1198701119860

2(119876)) Thus 119864[120587

1119860(119876)] is strictly





1119860(119876)]119889119876 = 0) Therefore 119864[120587

1119860(119876)]


lowast= 2119887119892(119886 minus 119887ℎ) is


lowast= 1199011119860

lowast(1198761119860

lowast) =

(119886 + 119887ℎ)2119887


119860(119901) = 119886 minus 119887119901





120578 (119901) = 1199061198871199012minus 119906 (119886 + 119887ℎ) 119901 + 119906ℎ119886 + 119892 lt 0 (A16)


lt 119901 lt

1199011198673119860


Ψ3(119901) = 1 minus 119865[

1198763119860

lowast(119901)

119911119860(119901)

]

= minus119892


119906

(119906 minus 119897)]

(A17)



119889Ψ (119901)

119889119901=

119892

(119906 minus 119897) (119901 minus ℎ)2

(119886 minus 119887119901)2(119886 + ℎ119887 minus 2119887119901) (A18)

1198892Ψ (119901)

1198891199012

= minus2119892

(119906 minus 119897) (119901 minus ℎ)3

(119886 minus 119887119901)3


] lt 0

(A19)

for all 1199011198713119860

lt 119901 lt 1199011198673119860


for 1199011198713119860

lt 119901 lt 1199011198673119860


3119860

lowast= (119886 + ℎ119887)2119887


lt 1199013119860

lowastlt 1199011198673119860


3119860

lowast=

1198760(1199013119860

lowast) = 2119887119892(119886 minus 119887ℎ)


119860(119901) = 119886 minus 119887119901


1198713119860lt 119901 lt

1199011198673119860


119864 [119902 (119901)] = 119911119860(119901) int

119906

1198760(119901)119911119860(119901)

119909119891 (119909) 119889119909

= Γ (119901)

=(119886 minus 119887119901) 119906

2

2 (119906 minus 119897)minus

1198922

2 (119906 minus 119897) (119901 minus ℎ)2

(119886 minus 119887119901)

(A20)


119889Γ (119901)

119889119901=

[1198922(2119886 + 119887ℎ minus 3119887119901) minus 119887119906

2(119901 minus ℎ)

3

(119886 minus 119887119901)2

]

2 (119906 minus 119897) (119901 minus ℎ)3

(119886 minus 119887119901)2

(A21)

1198892Γ (119901)

1198891199012

=minus1198922

3 (119906 minus 119897) (119901 minus ℎ)4

(119886 minus 119887119901)3

times [ 181198872(119901 minus

(2119886 + 119887ℎ)

3119887)

2

+ (119886 minus 119887ℎ)2]


(A22)


1199014119860=arg119901

1198922(2119886 + 119887ℎ minus 3119887119901) minus 119887119906

2(119901 minus ℎ)

3

(119886 minus 119887119901)2

= 0

(A23)

If 1199011198713119860

lt 1199014119860lt 1199011198673119860

then 119901 = 1199014119860


lowast= 1198760(1199014) = 119892(119901

4minus ℎ)


1198673119860

1198760(119901)119911119860(119901) = lim

119901rarr1199011198713119860

1198760(119901)119911119860(119901) = 119906


1198673119860

120578(119901) = lim119901rarr119901

1198713119860

120578(119901) = 0


1198673119860

119864[119902(119901)] =

lim119901rarr119901

1198713119860

119864[119902(119901)] = 0 If 1199014119860

le 1199011198713119860

or 1199014119860

ge 1199011198673119860


lt 119901 lt 1199011198673119860



this case


119887119886


1119872

lowast(119876) = 119901

0(119876) = ℎ + 119892119876



1119872(119876)119889119876 =

[ℎ + 119892119876]119887minus1ℎ + 119892(1 minus 119887)119876119886


119864 [1205871119872(119876)]

= int

119906

1198701119872(119876)

([1199011119872

lowast

(119876) minus ℎ] 119909119911119872[1199011119872

lowast

(119876)] minus 119892) 119891 (119909) 119889119909

=119892 [119906 minus 119870

1119872(119876)]

(119906 minus 119897)[119886

119876(119892

119876+ ℎ)

minus119887

(119906 + 119870

1119872(119876)

2) minus 1]

(A24)



119889119864 [1205871119872(119876)]

119889119876

= minus119886119892119876minus3[ℎ +

119892

119876]

minus119887minus1

[1199062minus 1198701119898

2

(119876)]ℎ119876 minus (119887 minus 1) 119892

2 (119906 minus 119897)

(A25)


1119872(119876)] is


1119872(0) = 0 (as

1198701119872(119876) = 119876

1119887ℎ + 119892119876


119864[1205871119872(0)] = +infin


1)119892ℎ 119889119864[1205871119872(119876)]119889119876 = 0 for 119876 = (119887 minus 1)119892ℎ and





1119872(119876)119889119876 gt



119887119886ℎ


1119872(119876) gt 119906





lowast= 1199011119872

lowast(1198761119872

lowast) = 119887ℎ(119887 minus 1)


119872(119901) = 119886119901






1198703119872(119901) =

119892119901119887

119886 (119901 minus ℎ)lt 119906 (A26)

(b) and (c)We have 1198891198703119872(119901)119889119901 = (119892119901

119887119886(119901minusℎ)

2)[(119887minus1)119901minus


is given by

Ψ3119872(119901) = 1 minus 119865[

1198760(119901)

119911119872(119901)

] (A27)

If 119897 lt 1198703119872(119901) lt 119906

Ψ3119872(119901) = minus

119892

119886 (119906 minus 119897)(

119901119887

119901 minus ℎ) + (1 +

119897

119906 minus 119897) (A28)


119889Ψ3119872(119901)

119889119901= minus

119892119901119887minus1


If 119897 ge 1198703119872(119901) Ψ

3119872(119901)= 1 and if 119906 le 119870

3119872(119901) Ψ

3119872(119901) = 0

(note that (1198763119872




unique1199013119872119897

1015840 and unique1199013119872119906

1015840 such that (i)1199013119872119897

1015840gt 1199013119872119906

1015840gt

ℎ (ii) 1198703119872(1199013119872119897

1015840) = 119897 and (iii) 119870

3119872(1199013119872119906

1015840) = 119906 Observed


119897 lt 1198703119872(119901) lt 119906 Therefore Ψ


for all 1199013119872119906

1015840lt 119901 lt 119901

3119872119897

1015840 and Ψ3119872(119897) = 1 for all 119901 ge 119901

3119872119897

1015840


3119872119906

1015840)For 119887 gt 1 119889119870


1198891198703119872(119901)119889119901 = 0 for 119901 = ℎ119887(119887minus1) and 119889119870

3119872(119901)119889119901 gt 0 for



= ℎ119887(119887 minus 1) As 1198703119872(1199013119872) =



0(119901) 119901)


119887lt 119906 as


3119872119906

10158401015840 and 1199013119872119906

101584010158401015840 such that (i)ℎ lt 119901

3119872119906

10158401015840lt 1199013119872119906

101584010158401015840 (ii) 1198703119872(1199013119872119906

10158401015840) = 119870

3119872(1199013119872119906

101584010158401015840) = 119906

and (iii) 1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 If(119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))

119887gt 119897 then 119897 lt 119870

3119872(119901) lt 119906 for

all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119906


10158401015840 and1199013119872119897

101584010158401015840 such that (i) 1199013119872119906

10158401015840lt 1199013119872119897

10158401015840le 1199013119872119897

101584010158401015840lt 1199013119872119906

101584010158401015840(ii) 119870

3119872(1199013119872119897

10158401015840) = 119870

3119872(1199013119872119897

101584010158401015840) = 119897 (iii) 119897 lt 119870

3119872(119901) lt 119906

for all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 or 1199013119872119897

101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 and(iv)1198703119872(119901) le 119897 for all 119901

3119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840Next for 119887 gt 1 and 119897 lt 119870

3119872(119901) lt 119906 119889Ψ

3119872(119901)119889119901 gt 0 for







1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897


101584010158401015840lt

119901 lt 1199013119872119906

101584010158401015840 and Ψ3119872(119901) = 1 for all 119901

3119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840


119864 [1199024119872(119901)] = Γ

1(119901) = int

119906

1198703119872(119901)

119909119891 (119909) 119889119909

=1199062

2 (119906 minus 119897)minus

11989221199012119887

21198862(119906 minus 119897) (119901 minus ℎ)

2

(A30)

for 119897 lt 1198703119872(119901) lt 119906 where119870

3119872(119901) is given by (23)

119864 [1199024119872(119901)] =

119906 minus 119897

2 for 119870

3119872(119901) le 119897

119864 [1199024119872(119901)] = 0 for 119870

3119872(119901) ge 119906

(A31)


below respectively

119889Γ1(119901)

119889119901= minus

11989221199012119887minus1

1198862(119906 minus 119897) (119901 minus ℎ)

3[(119887 minus 1) 119901 minus 119887ℎ] (A32)






3119872(119901) lt

119906 implies 1199013119872119906

1015840lt 119901 lt 119901

3119872119897

1015840 and 1198703119872(119901) le 119897 implies 119901 ge

1199013119872119897

1015840 where 1198703119872(1199013119872119897

1015840) = 119897 and 119870

3119872(1199013119872119906

1015840) = 119906 (see the


119864[1199024119872(1199013119872119897

1015840)] = (119906 minus 119897)2 for all 119901

3119872119906

1015840lt 119901 lt 119901

3119872119897


3119872119897

1015840For 119887 gt 1 and 119897 lt 119870

3119872(119901) lt 119906 119889Γ

1(119901)119889119901 gt 0 for all





(119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))119887lt 119906


1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906


119887lt 119906

119897 lt 1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906


1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 or 1199013119872119897

101584010158401015840lt

119901 lt 1199013119872119906

101584010158401015840 and 1198703119872(119901) le 119897 for all 119901

3119872119897

10158401015840lt 119901 lt

1199013119872119897





for all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897


101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 and 119864[1199024119872(119901)] = (119906 minus 119897)2 for all

1199013119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840



Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119889119864 [120587 (119876)]

119889119876= minus

120572119891 [119876119911 (119901)] [(119901 minus ℎ)119876 minus 119892]

119911 (119901) (A5)

1198892119864 [120587 (119876)]

1198891198762

= minus120572((119901 minus ℎ) 119911 (119901) 119891[119876

119911 (119901)]

+ [(119901 minus ℎ)119876 minus 119892]1198911015840[

119876

119911 (119901)]) times (119911

2(119901))

minus1

(A6)


(119901 minus ℎ) 119911 (119901) 119891 [119876

119911 (119901)] + [(119901 minus ℎ)119876 minus 119892]119891

1015840[

119876

119911 (119901)]

ge (119901 minus ℎ) 119911 (119901) 119891 [119876

119911 (119901)] gt 0

(A7)




3


is given by

Ψ (119901) = 1 minus 119865[1198763

lowast(119901)

119911 (119901)] = 1 minus 119865[

119892

(119901 minus ℎ) 119911 (119901)] (A8)


119889Ψ (119901)

119889119901= [

119892

(119901 minus ℎ)2

1199112(119901)

] [119911 (119901) + (119901 minus ℎ) 1199111015840(119901)]

times 119891[119892

(119901 minus ℎ) 119911 (119901)]

(A9)



119911(1199013

lowast) + (119901

3


1015840(1199013


(A9)



4




is in the form

119889119864 [119902 (119901)]

119889119901=

120577 (119901)

[(119901 minus ℎ)3

1199112(119901)]

(A10)


4


4






1119860(119876) where 119870

1119860(119876) =

119876119911119860(1199011119860


1119860(119876) lt 119906


1198762minus (119886 minus 119887ℎ) 119906119876 + 119887119892119906

gt 1198762minus (119886 minus 119887ℎ) 119906119876 +

(119886 minus 119887ℎ)21199062

4

= [119876 minus(119886 minus 119887ℎ) 119906

2]

2

ge 0

(A11)


given by

1205871119860(119876)

=

[1199011119860

lowast(119876) minus ℎ] 119909119911

119860[1199011119860


1119860(119876)

0 ow(A12)

where 1199011119860

lowast(119876) = 119901



119864 [1205871119860(119876)]

= int

119906

1198701119860(119876)

([1199011119860

lowast

(119876) minus ℎ] 119909119911119860[1199011119860

lowast

(119876)] minus 119892) 119891 (119909) 119889119909

=119892 [119906 minus 119870

1119860(119876)]

(119906 minus 119897)([119886 minus 119887ℎ

119876minus119887119892

1198762] [

119906 + 1198701119860(119876)

2] minus 1)

(A13)



119889119864 [1205871119860(119876)]

119889119876=

119892 [1199062minus 1198701119860

2(119876)] [minus (119886 minus 119887ℎ)119876 + 2119887119892]

2 (119906 minus 119897) 1198763

(A14)

1198892119864 [1205871119860(119876)]

1198891198762

=

119892 [1199062minus 1198701119860

2(119876)] [(119886 minus 119887ℎ)119876 minus 3119887119892]

(119906 minus 119897) 1198764

(A15)

From (A15) for 119876 gt 0 1198892119864[1205871119860(119876)]119889119876


3119887119892(119886 minus 119887ℎ) (as 1199062 gt 1198701119860

2(119876)) Thus 119864[120587

1119860(119876)] is strictly





1119860(119876)]119889119876 = 0) Therefore 119864[120587

1119860(119876)]


lowast= 2119887119892(119886 minus 119887ℎ) is


lowast= 1199011119860

lowast(1198761119860

lowast) =

(119886 + 119887ℎ)2119887


119860(119901) = 119886 minus 119887119901





120578 (119901) = 1199061198871199012minus 119906 (119886 + 119887ℎ) 119901 + 119906ℎ119886 + 119892 lt 0 (A16)


lt 119901 lt

1199011198673119860


Ψ3(119901) = 1 minus 119865[

1198763119860

lowast(119901)

119911119860(119901)

]

= minus119892


119906

(119906 minus 119897)]

(A17)



119889Ψ (119901)

119889119901=

119892

(119906 minus 119897) (119901 minus ℎ)2

(119886 minus 119887119901)2(119886 + ℎ119887 minus 2119887119901) (A18)

1198892Ψ (119901)

1198891199012

= minus2119892

(119906 minus 119897) (119901 minus ℎ)3

(119886 minus 119887119901)3


] lt 0

(A19)

for all 1199011198713119860

lt 119901 lt 1199011198673119860


for 1199011198713119860

lt 119901 lt 1199011198673119860


3119860

lowast= (119886 + ℎ119887)2119887


lt 1199013119860

lowastlt 1199011198673119860


3119860

lowast=

1198760(1199013119860

lowast) = 2119887119892(119886 minus 119887ℎ)


119860(119901) = 119886 minus 119887119901


1198713119860lt 119901 lt

1199011198673119860


119864 [119902 (119901)] = 119911119860(119901) int

119906

1198760(119901)119911119860(119901)

119909119891 (119909) 119889119909

= Γ (119901)

=(119886 minus 119887119901) 119906

2

2 (119906 minus 119897)minus

1198922

2 (119906 minus 119897) (119901 minus ℎ)2

(119886 minus 119887119901)

(A20)


119889Γ (119901)

119889119901=

[1198922(2119886 + 119887ℎ minus 3119887119901) minus 119887119906

2(119901 minus ℎ)

3

(119886 minus 119887119901)2

]

2 (119906 minus 119897) (119901 minus ℎ)3

(119886 minus 119887119901)2

(A21)

1198892Γ (119901)

1198891199012

=minus1198922

3 (119906 minus 119897) (119901 minus ℎ)4

(119886 minus 119887119901)3

times [ 181198872(119901 minus

(2119886 + 119887ℎ)

3119887)

2

+ (119886 minus 119887ℎ)2]


(A22)


1199014119860=arg119901

1198922(2119886 + 119887ℎ minus 3119887119901) minus 119887119906

2(119901 minus ℎ)

3

(119886 minus 119887119901)2

= 0

(A23)

If 1199011198713119860

lt 1199014119860lt 1199011198673119860

then 119901 = 1199014119860


lowast= 1198760(1199014) = 119892(119901

4minus ℎ)


1198673119860

1198760(119901)119911119860(119901) = lim

119901rarr1199011198713119860

1198760(119901)119911119860(119901) = 119906


1198673119860

120578(119901) = lim119901rarr119901

1198713119860

120578(119901) = 0


1198673119860

119864[119902(119901)] =

lim119901rarr119901

1198713119860

119864[119902(119901)] = 0 If 1199014119860

le 1199011198713119860

or 1199014119860

ge 1199011198673119860


lt 119901 lt 1199011198673119860



this case


119887119886


1119872

lowast(119876) = 119901

0(119876) = ℎ + 119892119876



1119872(119876)119889119876 =

[ℎ + 119892119876]119887minus1ℎ + 119892(1 minus 119887)119876119886


119864 [1205871119872(119876)]

= int

119906

1198701119872(119876)

([1199011119872

lowast

(119876) minus ℎ] 119909119911119872[1199011119872

lowast

(119876)] minus 119892) 119891 (119909) 119889119909

=119892 [119906 minus 119870

1119872(119876)]

(119906 minus 119897)[119886

119876(119892

119876+ ℎ)

minus119887

(119906 + 119870

1119872(119876)

2) minus 1]

(A24)



119889119864 [1205871119872(119876)]

119889119876

= minus119886119892119876minus3[ℎ +

119892

119876]

minus119887minus1

[1199062minus 1198701119898

2

(119876)]ℎ119876 minus (119887 minus 1) 119892

2 (119906 minus 119897)

(A25)


1119872(119876)] is


1119872(0) = 0 (as

1198701119872(119876) = 119876

1119887ℎ + 119892119876


119864[1205871119872(0)] = +infin


1)119892ℎ 119889119864[1205871119872(119876)]119889119876 = 0 for 119876 = (119887 minus 1)119892ℎ and





1119872(119876)119889119876 gt



119887119886ℎ


1119872(119876) gt 119906





lowast= 1199011119872

lowast(1198761119872

lowast) = 119887ℎ(119887 minus 1)


119872(119901) = 119886119901






1198703119872(119901) =

119892119901119887

119886 (119901 minus ℎ)lt 119906 (A26)

(b) and (c)We have 1198891198703119872(119901)119889119901 = (119892119901

119887119886(119901minusℎ)

2)[(119887minus1)119901minus


is given by

Ψ3119872(119901) = 1 minus 119865[

1198760(119901)

119911119872(119901)

] (A27)

If 119897 lt 1198703119872(119901) lt 119906

Ψ3119872(119901) = minus

119892

119886 (119906 minus 119897)(

119901119887

119901 minus ℎ) + (1 +

119897

119906 minus 119897) (A28)


119889Ψ3119872(119901)

119889119901= minus

119892119901119887minus1


If 119897 ge 1198703119872(119901) Ψ

3119872(119901)= 1 and if 119906 le 119870

3119872(119901) Ψ

3119872(119901) = 0

(note that (1198763119872




unique1199013119872119897

1015840 and unique1199013119872119906

1015840 such that (i)1199013119872119897

1015840gt 1199013119872119906

1015840gt

ℎ (ii) 1198703119872(1199013119872119897

1015840) = 119897 and (iii) 119870

3119872(1199013119872119906

1015840) = 119906 Observed


119897 lt 1198703119872(119901) lt 119906 Therefore Ψ


for all 1199013119872119906

1015840lt 119901 lt 119901

3119872119897

1015840 and Ψ3119872(119897) = 1 for all 119901 ge 119901

3119872119897

1015840


3119872119906

1015840)For 119887 gt 1 119889119870


1198891198703119872(119901)119889119901 = 0 for 119901 = ℎ119887(119887minus1) and 119889119870

3119872(119901)119889119901 gt 0 for



= ℎ119887(119887 minus 1) As 1198703119872(1199013119872) =



0(119901) 119901)


119887lt 119906 as


3119872119906

10158401015840 and 1199013119872119906

101584010158401015840 such that (i)ℎ lt 119901

3119872119906

10158401015840lt 1199013119872119906

101584010158401015840 (ii) 1198703119872(1199013119872119906

10158401015840) = 119870

3119872(1199013119872119906

101584010158401015840) = 119906

and (iii) 1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 If(119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))

119887gt 119897 then 119897 lt 119870

3119872(119901) lt 119906 for

all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119906


10158401015840 and1199013119872119897

101584010158401015840 such that (i) 1199013119872119906

10158401015840lt 1199013119872119897

10158401015840le 1199013119872119897

101584010158401015840lt 1199013119872119906

101584010158401015840(ii) 119870

3119872(1199013119872119897

10158401015840) = 119870

3119872(1199013119872119897

101584010158401015840) = 119897 (iii) 119897 lt 119870

3119872(119901) lt 119906

for all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 or 1199013119872119897

101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 and(iv)1198703119872(119901) le 119897 for all 119901

3119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840Next for 119887 gt 1 and 119897 lt 119870

3119872(119901) lt 119906 119889Ψ

3119872(119901)119889119901 gt 0 for







1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897


101584010158401015840lt

119901 lt 1199013119872119906

101584010158401015840 and Ψ3119872(119901) = 1 for all 119901

3119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840


119864 [1199024119872(119901)] = Γ

1(119901) = int

119906

1198703119872(119901)

119909119891 (119909) 119889119909

=1199062

2 (119906 minus 119897)minus

11989221199012119887

21198862(119906 minus 119897) (119901 minus ℎ)

2

(A30)

for 119897 lt 1198703119872(119901) lt 119906 where119870

3119872(119901) is given by (23)

119864 [1199024119872(119901)] =

119906 minus 119897

2 for 119870

3119872(119901) le 119897

119864 [1199024119872(119901)] = 0 for 119870

3119872(119901) ge 119906

(A31)


below respectively

119889Γ1(119901)

119889119901= minus

11989221199012119887minus1

1198862(119906 minus 119897) (119901 minus ℎ)

3[(119887 minus 1) 119901 minus 119887ℎ] (A32)






3119872(119901) lt

119906 implies 1199013119872119906

1015840lt 119901 lt 119901

3119872119897

1015840 and 1198703119872(119901) le 119897 implies 119901 ge

1199013119872119897

1015840 where 1198703119872(1199013119872119897

1015840) = 119897 and 119870

3119872(1199013119872119906

1015840) = 119906 (see the


119864[1199024119872(1199013119872119897

1015840)] = (119906 minus 119897)2 for all 119901

3119872119906

1015840lt 119901 lt 119901

3119872119897


3119872119897

1015840For 119887 gt 1 and 119897 lt 119870

3119872(119901) lt 119906 119889Γ

1(119901)119889119901 gt 0 for all





(119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))119887lt 119906


1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906


119887lt 119906

119897 lt 1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906


1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 or 1199013119872119897

101584010158401015840lt

119901 lt 1199013119872119906

101584010158401015840 and 1198703119872(119901) le 119897 for all 119901

3119872119897

10158401015840lt 119901 lt

1199013119872119897





for all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897


101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 and 119864[1199024119872(119901)] = (119906 minus 119897)2 for all

1199013119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840



Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1119860(119876)]119889119876 = 0) Therefore 119864[120587

1119860(119876)]


lowast= 2119887119892(119886 minus 119887ℎ) is


lowast= 1199011119860

lowast(1198761119860

lowast) =

(119886 + 119887ℎ)2119887


119860(119901) = 119886 minus 119887119901





120578 (119901) = 1199061198871199012minus 119906 (119886 + 119887ℎ) 119901 + 119906ℎ119886 + 119892 lt 0 (A16)


lt 119901 lt

1199011198673119860


Ψ3(119901) = 1 minus 119865[

1198763119860

lowast(119901)

119911119860(119901)

]

= minus119892


119906

(119906 minus 119897)]

(A17)



119889Ψ (119901)

119889119901=

119892

(119906 minus 119897) (119901 minus ℎ)2

(119886 minus 119887119901)2(119886 + ℎ119887 minus 2119887119901) (A18)

1198892Ψ (119901)

1198891199012

= minus2119892

(119906 minus 119897) (119901 minus ℎ)3

(119886 minus 119887119901)3


] lt 0

(A19)

for all 1199011198713119860

lt 119901 lt 1199011198673119860


for 1199011198713119860

lt 119901 lt 1199011198673119860


3119860

lowast= (119886 + ℎ119887)2119887


lt 1199013119860

lowastlt 1199011198673119860


3119860

lowast=

1198760(1199013119860

lowast) = 2119887119892(119886 minus 119887ℎ)


119860(119901) = 119886 minus 119887119901


1198713119860lt 119901 lt

1199011198673119860


119864 [119902 (119901)] = 119911119860(119901) int

119906

1198760(119901)119911119860(119901)

119909119891 (119909) 119889119909

= Γ (119901)

=(119886 minus 119887119901) 119906

2

2 (119906 minus 119897)minus

1198922

2 (119906 minus 119897) (119901 minus ℎ)2

(119886 minus 119887119901)

(A20)


119889Γ (119901)

119889119901=

[1198922(2119886 + 119887ℎ minus 3119887119901) minus 119887119906

2(119901 minus ℎ)

3

(119886 minus 119887119901)2

]

2 (119906 minus 119897) (119901 minus ℎ)3

(119886 minus 119887119901)2

(A21)

1198892Γ (119901)

1198891199012

=minus1198922

3 (119906 minus 119897) (119901 minus ℎ)4

(119886 minus 119887119901)3

times [ 181198872(119901 minus

(2119886 + 119887ℎ)

3119887)

2

+ (119886 minus 119887ℎ)2]


(A22)


1199014119860=arg119901

1198922(2119886 + 119887ℎ minus 3119887119901) minus 119887119906

2(119901 minus ℎ)

3

(119886 minus 119887119901)2

= 0

(A23)

If 1199011198713119860

lt 1199014119860lt 1199011198673119860

then 119901 = 1199014119860


lowast= 1198760(1199014) = 119892(119901

4minus ℎ)


1198673119860

1198760(119901)119911119860(119901) = lim

119901rarr1199011198713119860

1198760(119901)119911119860(119901) = 119906


1198673119860

120578(119901) = lim119901rarr119901

1198713119860

120578(119901) = 0


1198673119860

119864[119902(119901)] =

lim119901rarr119901

1198713119860

119864[119902(119901)] = 0 If 1199014119860

le 1199011198713119860

or 1199014119860

ge 1199011198673119860


lt 119901 lt 1199011198673119860



this case


119887119886


1119872

lowast(119876) = 119901

0(119876) = ℎ + 119892119876



1119872(119876)119889119876 =

[ℎ + 119892119876]119887minus1ℎ + 119892(1 minus 119887)119876119886


119864 [1205871119872(119876)]

= int

119906

1198701119872(119876)

([1199011119872

lowast

(119876) minus ℎ] 119909119911119872[1199011119872

lowast

(119876)] minus 119892) 119891 (119909) 119889119909

=119892 [119906 minus 119870

1119872(119876)]

(119906 minus 119897)[119886

119876(119892

119876+ ℎ)

minus119887

(119906 + 119870

1119872(119876)

2) minus 1]

(A24)



119889119864 [1205871119872(119876)]

119889119876

= minus119886119892119876minus3[ℎ +

119892

119876]

minus119887minus1

[1199062minus 1198701119898

2

(119876)]ℎ119876 minus (119887 minus 1) 119892

2 (119906 minus 119897)

(A25)


1119872(119876)] is


1119872(0) = 0 (as

1198701119872(119876) = 119876

1119887ℎ + 119892119876


119864[1205871119872(0)] = +infin


1)119892ℎ 119889119864[1205871119872(119876)]119889119876 = 0 for 119876 = (119887 minus 1)119892ℎ and





1119872(119876)119889119876 gt



119887119886ℎ


1119872(119876) gt 119906





lowast= 1199011119872

lowast(1198761119872

lowast) = 119887ℎ(119887 minus 1)


119872(119901) = 119886119901






1198703119872(119901) =

119892119901119887

119886 (119901 minus ℎ)lt 119906 (A26)

(b) and (c)We have 1198891198703119872(119901)119889119901 = (119892119901

119887119886(119901minusℎ)

2)[(119887minus1)119901minus


is given by

Ψ3119872(119901) = 1 minus 119865[

1198760(119901)

119911119872(119901)

] (A27)

If 119897 lt 1198703119872(119901) lt 119906

Ψ3119872(119901) = minus

119892

119886 (119906 minus 119897)(

119901119887

119901 minus ℎ) + (1 +

119897

119906 minus 119897) (A28)


119889Ψ3119872(119901)

119889119901= minus

119892119901119887minus1


If 119897 ge 1198703119872(119901) Ψ

3119872(119901)= 1 and if 119906 le 119870

3119872(119901) Ψ

3119872(119901) = 0

(note that (1198763119872




unique1199013119872119897

1015840 and unique1199013119872119906

1015840 such that (i)1199013119872119897

1015840gt 1199013119872119906

1015840gt

ℎ (ii) 1198703119872(1199013119872119897

1015840) = 119897 and (iii) 119870

3119872(1199013119872119906

1015840) = 119906 Observed


119897 lt 1198703119872(119901) lt 119906 Therefore Ψ


for all 1199013119872119906

1015840lt 119901 lt 119901

3119872119897

1015840 and Ψ3119872(119897) = 1 for all 119901 ge 119901

3119872119897

1015840


3119872119906

1015840)For 119887 gt 1 119889119870


1198891198703119872(119901)119889119901 = 0 for 119901 = ℎ119887(119887minus1) and 119889119870

3119872(119901)119889119901 gt 0 for



= ℎ119887(119887 minus 1) As 1198703119872(1199013119872) =



0(119901) 119901)


119887lt 119906 as


3119872119906

10158401015840 and 1199013119872119906

101584010158401015840 such that (i)ℎ lt 119901

3119872119906

10158401015840lt 1199013119872119906

101584010158401015840 (ii) 1198703119872(1199013119872119906

10158401015840) = 119870

3119872(1199013119872119906

101584010158401015840) = 119906

and (iii) 1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 If(119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))

119887gt 119897 then 119897 lt 119870

3119872(119901) lt 119906 for

all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119906


10158401015840 and1199013119872119897

101584010158401015840 such that (i) 1199013119872119906

10158401015840lt 1199013119872119897

10158401015840le 1199013119872119897

101584010158401015840lt 1199013119872119906

101584010158401015840(ii) 119870

3119872(1199013119872119897

10158401015840) = 119870

3119872(1199013119872119897

101584010158401015840) = 119897 (iii) 119897 lt 119870

3119872(119901) lt 119906

for all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 or 1199013119872119897

101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 and(iv)1198703119872(119901) le 119897 for all 119901

3119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840Next for 119887 gt 1 and 119897 lt 119870

3119872(119901) lt 119906 119889Ψ

3119872(119901)119889119901 gt 0 for







1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897


101584010158401015840lt

119901 lt 1199013119872119906

101584010158401015840 and Ψ3119872(119901) = 1 for all 119901

3119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840


119864 [1199024119872(119901)] = Γ

1(119901) = int

119906

1198703119872(119901)

119909119891 (119909) 119889119909

=1199062

2 (119906 minus 119897)minus

11989221199012119887

21198862(119906 minus 119897) (119901 minus ℎ)

2

(A30)

for 119897 lt 1198703119872(119901) lt 119906 where119870

3119872(119901) is given by (23)

119864 [1199024119872(119901)] =

119906 minus 119897

2 for 119870

3119872(119901) le 119897

119864 [1199024119872(119901)] = 0 for 119870

3119872(119901) ge 119906

(A31)


below respectively

119889Γ1(119901)

119889119901= minus

11989221199012119887minus1

1198862(119906 minus 119897) (119901 minus ℎ)

3[(119887 minus 1) 119901 minus 119887ℎ] (A32)






3119872(119901) lt

119906 implies 1199013119872119906

1015840lt 119901 lt 119901

3119872119897

1015840 and 1198703119872(119901) le 119897 implies 119901 ge

1199013119872119897

1015840 where 1198703119872(1199013119872119897

1015840) = 119897 and 119870

3119872(1199013119872119906

1015840) = 119906 (see the


119864[1199024119872(1199013119872119897

1015840)] = (119906 minus 119897)2 for all 119901

3119872119906

1015840lt 119901 lt 119901

3119872119897


3119872119897

1015840For 119887 gt 1 and 119897 lt 119870

3119872(119901) lt 119906 119889Γ

1(119901)119889119901 gt 0 for all





(119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))119887lt 119906


1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906


119887lt 119906

119897 lt 1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906


1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 or 1199013119872119897

101584010158401015840lt

119901 lt 1199013119872119906

101584010158401015840 and 1198703119872(119901) le 119897 for all 119901

3119872119897

10158401015840lt 119901 lt

1199013119872119897





for all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897


101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 and 119864[1199024119872(119901)] = (119906 minus 119897)2 for all

1199013119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840



Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119889119864 [1205871119872(119876)]

119889119876

= minus119886119892119876minus3[ℎ +

119892

119876]

minus119887minus1

[1199062minus 1198701119898

2

(119876)]ℎ119876 minus (119887 minus 1) 119892

2 (119906 minus 119897)

(A25)


1119872(119876)] is


1119872(0) = 0 (as

1198701119872(119876) = 119876

1119887ℎ + 119892119876


119864[1205871119872(0)] = +infin


1)119892ℎ 119889119864[1205871119872(119876)]119889119876 = 0 for 119876 = (119887 minus 1)119892ℎ and





1119872(119876)119889119876 gt



119887119886ℎ


1119872(119876) gt 119906





lowast= 1199011119872

lowast(1198761119872

lowast) = 119887ℎ(119887 minus 1)


119872(119901) = 119886119901






1198703119872(119901) =

119892119901119887

119886 (119901 minus ℎ)lt 119906 (A26)

(b) and (c)We have 1198891198703119872(119901)119889119901 = (119892119901

119887119886(119901minusℎ)

2)[(119887minus1)119901minus


is given by

Ψ3119872(119901) = 1 minus 119865[

1198760(119901)

119911119872(119901)

] (A27)

If 119897 lt 1198703119872(119901) lt 119906

Ψ3119872(119901) = minus

119892

119886 (119906 minus 119897)(

119901119887

119901 minus ℎ) + (1 +

119897

119906 minus 119897) (A28)


119889Ψ3119872(119901)

119889119901= minus

119892119901119887minus1


If 119897 ge 1198703119872(119901) Ψ

3119872(119901)= 1 and if 119906 le 119870

3119872(119901) Ψ

3119872(119901) = 0

(note that (1198763119872




unique1199013119872119897

1015840 and unique1199013119872119906

1015840 such that (i)1199013119872119897

1015840gt 1199013119872119906

1015840gt

ℎ (ii) 1198703119872(1199013119872119897

1015840) = 119897 and (iii) 119870

3119872(1199013119872119906

1015840) = 119906 Observed


119897 lt 1198703119872(119901) lt 119906 Therefore Ψ


for all 1199013119872119906

1015840lt 119901 lt 119901

3119872119897

1015840 and Ψ3119872(119897) = 1 for all 119901 ge 119901

3119872119897

1015840


3119872119906

1015840)For 119887 gt 1 119889119870


1198891198703119872(119901)119889119901 = 0 for 119901 = ℎ119887(119887minus1) and 119889119870

3119872(119901)119889119901 gt 0 for



= ℎ119887(119887 minus 1) As 1198703119872(1199013119872) =



0(119901) 119901)


119887lt 119906 as


3119872119906

10158401015840 and 1199013119872119906

101584010158401015840 such that (i)ℎ lt 119901

3119872119906

10158401015840lt 1199013119872119906

101584010158401015840 (ii) 1198703119872(1199013119872119906

10158401015840) = 119870

3119872(1199013119872119906

101584010158401015840) = 119906

and (iii) 1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 If(119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))

119887gt 119897 then 119897 lt 119870

3119872(119901) lt 119906 for

all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119906


10158401015840 and1199013119872119897

101584010158401015840 such that (i) 1199013119872119906

10158401015840lt 1199013119872119897

10158401015840le 1199013119872119897

101584010158401015840lt 1199013119872119906

101584010158401015840(ii) 119870

3119872(1199013119872119897

10158401015840) = 119870

3119872(1199013119872119897

101584010158401015840) = 119897 (iii) 119897 lt 119870

3119872(119901) lt 119906

for all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 or 1199013119872119897

101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 and(iv)1198703119872(119901) le 119897 for all 119901

3119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840Next for 119887 gt 1 and 119897 lt 119870

3119872(119901) lt 119906 119889Ψ

3119872(119901)119889119901 gt 0 for







1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897


101584010158401015840lt

119901 lt 1199013119872119906

101584010158401015840 and Ψ3119872(119901) = 1 for all 119901

3119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840


119864 [1199024119872(119901)] = Γ

1(119901) = int

119906

1198703119872(119901)

119909119891 (119909) 119889119909

=1199062

2 (119906 minus 119897)minus

11989221199012119887

21198862(119906 minus 119897) (119901 minus ℎ)

2

(A30)

for 119897 lt 1198703119872(119901) lt 119906 where119870

3119872(119901) is given by (23)

119864 [1199024119872(119901)] =

119906 minus 119897

2 for 119870

3119872(119901) le 119897

119864 [1199024119872(119901)] = 0 for 119870

3119872(119901) ge 119906

(A31)


below respectively

119889Γ1(119901)

119889119901= minus

11989221199012119887minus1

1198862(119906 minus 119897) (119901 minus ℎ)

3[(119887 minus 1) 119901 minus 119887ℎ] (A32)






3119872(119901) lt

119906 implies 1199013119872119906

1015840lt 119901 lt 119901

3119872119897

1015840 and 1198703119872(119901) le 119897 implies 119901 ge

1199013119872119897

1015840 where 1198703119872(1199013119872119897

1015840) = 119897 and 119870

3119872(1199013119872119906

1015840) = 119906 (see the


119864[1199024119872(1199013119872119897

1015840)] = (119906 minus 119897)2 for all 119901

3119872119906

1015840lt 119901 lt 119901

3119872119897


3119872119897

1015840For 119887 gt 1 and 119897 lt 119870

3119872(119901) lt 119906 119889Γ

1(119901)119889119901 gt 0 for all





(119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))119887lt 119906


1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906


119887lt 119906

119897 lt 1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906


1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 or 1199013119872119897

101584010158401015840lt

119901 lt 1199013119872119906

101584010158401015840 and 1198703119872(119901) le 119897 for all 119901

3119872119897

10158401015840lt 119901 lt

1199013119872119897





for all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897


101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 and 119864[1199024119872(119901)] = (119906 minus 119897)2 for all

1199013119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840



Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











3119872(119901) lt

119906 implies 1199013119872119906

1015840lt 119901 lt 119901

3119872119897

1015840 and 1198703119872(119901) le 119897 implies 119901 ge

1199013119872119897

1015840 where 1198703119872(1199013119872119897

1015840) = 119897 and 119870

3119872(1199013119872119906

1015840) = 119906 (see the


119864[1199024119872(1199013119872119897

1015840)] = (119906 minus 119897)2 for all 119901

3119872119906

1015840lt 119901 lt 119901

3119872119897


3119872119897

1015840For 119887 gt 1 and 119897 lt 119870

3119872(119901) lt 119906 119889Γ

1(119901)119889119901 gt 0 for all





(119892(119887 minus 1)119886ℎ)(ℎ119887(119887 minus 1))119887lt 119906


1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906


119887lt 119906

119897 lt 1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119906


1198703119872(119901) lt 119906 for all 119901

3119872119906

10158401015840lt 119901 lt 119901

3119872119897

10158401015840 or 1199013119872119897

101584010158401015840lt

119901 lt 1199013119872119906

101584010158401015840 and 1198703119872(119901) le 119897 for all 119901

3119872119897

10158401015840lt 119901 lt

1199013119872119897





for all 1199013119872119906

10158401015840lt 119901 lt 119901

3119872119897


101584010158401015840lt 119901 lt 119901

3119872119906

101584010158401015840 and 119864[1199024119872(119901)] = (119906 minus 119897)2 for all

1199013119872119897

10158401015840lt 119901 lt 119901

3119872119897

101584010158401015840



Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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