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Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 127615 7 pageshttpdxdoiorg1011552013127615
Research ArticleFekete-Szegy Inequality for a Subclass of119901-Valent Analytic Functions
Mohsan Raza1 Muhammad Arif2 and Maslina Darus3
1 Department of Mathematics Government College University Faisalabad Faisalabad Punjab 38000 Pakistan2Department of Mathematics Abdul Wali Khan University Mardan Mardan Khyber Pakhtunkhwa 23200 Pakistan3 School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia 43600 BangiSelangor D Ehsan Malaysia
Correspondence should be addressed to Muhammad Arif marifmathsyahoocom
Received 18 January 2013 Accepted 11 April 2013
Academic Editor Mina Abd-El-Malek
Copyright copy 2013 Mohsan Raza et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Themain object of this paper is to study Fekete-Szego problem for the class of 119901-valent functions Fekete-Szego inequality of severalclasses is obtained as special cases from our results Applications of the results are also obtained on the class defined by convolution
1 Introduction and Preliminaries
Let 119860119901 denote the class of functions 119891(119911) of the form
119891 (119911) = 119911119901+
infin
sum
119899=119901+1
119886119899119911119899 (119901 isin N = 1 2 3 ) (1)
which are analytic in the open unit disk 119864 Also 1198601 = 119860 theusual class of analytic functions defined in the open unit disk119864 = 119911 |119911| lt 1 Let 119891(119911) and 119892(119911) be analytic in 119864 We saythat the function 119891 is subordinate to the function 119892 and write119891(119911) ≺ 119892(119911) if and only if there exists Schwarz function 119908analytic in 119864 such that 119908(0) = 0 |119908(119911)| lt 1 for 119911 isin 119864 and119891(119911) = 119892(119908(119911)) In particular if 119892 is univalent in 119864 then wehave the following equivalence
119891 (119911) ≺ 119892 (119911) lArrrArr 119891 (0) = 119892 (0) 119891 (119864) sub 119892 (119864) (2)
For any two analytic functions 119891(119911) of the form (1) and119892(119911) with
119892 (119911) = 119911119901+
infin
sum
119899=119901+1
119887119899119911119899 119911 isin 119864 (3)
the convolution (Hadamard product) is given by
(119891 lowast 119892) (119911) = 119911119901+
infin
sum
119899=119901+1
119886119899119887119899119911119899 119911 isin 119864 (4)
Let 120601(119911) be an analytic function with positive real part on 119864with 120601(0) = 1 1206011015840(0) gt 0 which maps the unit disk 119864 ontoa region starlike with respect to 1 which is symmetric withrespect to the real axis Denote by 119878lowast
119901(120601) the class of functions
119891 analytic in 119864 for which
1199111198911015840(119911)
119901119891 (119911)≺ 120601 (119911) 119911 isin 119864 (5)
The class 119878lowast
119901(120601)was defined and studied by Ali et al [1]They
obtained the Fekete-Szego inequality for functions in the class119878lowast
119901(120601)The class 119878lowast
1(120601) coincideswith the class 119878lowast(120601) discussed
by Ma and Minda [2] Owa [3] introduced a subclass of 119901-valently Bazilevic functions 119867119901(119860 119861 120572 120573) A function 119891 isin
119860119901 is said to be in the class119867119901(119860 119861 120572 120573) if and only if
(1 minus 120573) (119891(119911)
119911119901)
120572
+ 1205731199111198911015840(119911)
119901119891 (119911)(119891 (119911)
119911119901)
120572
≺1 + 119860119911
1 + 119861119911 119911 isin 119864
(6)
where minus1 le 119861 lt 119860 le 1 120572 ge 0 and 0 le 120573 le 1 We now definethe following subclass of analytic functions
Definition 1 Let 120601(119911) be a univalent starlike function withrespect to 1 which maps the unit disk 119864 onto a region in theright half-plane which is symmetric with respect to the real
2 Journal of Applied Mathematics
axis with 120601(0) = 1 and 1206011015840(0) gt 0 A function 119891 isin 119860119901 is in theclass 119881119901119887120572120573(120601) if
1 minus2
119887+2
119887 (1 minus 120573) (
119891 (119911)
119911119901)
120572
+ 1205731199111198911015840(119911)
119901119891 (119911)(119891 (119911)
119911119901)
120572
≺ 120601 (119911)
(7)
where 0 le 120573 le 1 120572 ge 0 and 119887 gt 0
Definition 2 A function 119891 isin 119860119901 is in the class 119881119901119887120572120573119892(120601) if
1 minus2
119887+2
119887 (1 minus 120573)(
(119891 lowast 119892) (119911)
119911119901)
120572
+ 120573119911(119891 lowast 119892)
1015840(119911)
119901 (119891 lowast 119892) (119911)((119891 lowast 119892) (119911)
119911119901)
120572
≺ 120601 (119911)
(8)
where 0 le 120573 le 1 120572 ge 0 and 119887 gt 0 In other words a func-tion 119891 isin 119860119901 is in the class 119881119901119887120572120573119892(120601) if (119891 lowast 119892)(119911) isin
119881119901119887120572120573(120601)We have the following special cases
(i) 119881119901211(120601) coincides with the class 119878lowast119901(120601) introduced
and studied by Ali et al [1](ii) For 119901 = 1 119887 = 2 and 120573 = 1 we have the class 119861120572(120601)
introduced and studied by Ravichandran et al [4](iii) For 119887 = 2 and 120601(119911) = (1 + 119860119911)(1 + 119861119911) the class
1198811199012120572120573(120601) reduces to 119867119901(119860 119861 120572 120573) introduced andstudied by Owa [3]
(iv) For 120601(119911) = (1 + (1 minus 2120574)119911)(1 minus 119911) the class 11988111990121205721(120601)reduces to the class 119861119901(120572 120574) defined as
119861119901 (120572 120574) = 119891 isin 119860119901 Re(1199111198911015840(119911)
119901119891 (119911)(119891 (119911)
119911119901)
120572
) gt 120574
0 le 120574 lt 1 119911 isin 119864
(9)
(v) For 120601(119911) = (1 + (1 minus 2120574)119911)(1 minus 119911) the class 11988111990121205720(120601)is defined as
119891 isin 119860119901 Re(119891 (119911)
119911119901)
120572
gt 120574 0 le 120574 lt 1 119911 isin 119864 (10)
(vi) 1198811201(120601) = 119878lowast(120601) is investigated by Ma and Minda
[2](vii) For 120601(119911) = (1 + (1 minus 2120574)119911)(1 minus 119911) the class 1198811210(120601)
reduces to the class
119861120574 = 119891 isin 1198601 Re119891 (119911)
119911gt 120574 (11)
studied by Chen [5]
We need the following results to obtain our main results
Lemma 3 (see [1]) LetΩ be the class of analytic functions119908normalized by119908(0) = 0 satisfying condition |119908(119911)| lt 1 If119908 isin
Ω and 119908(119911) = 1199081119911 + 11990821199112 + sdot sdot sdot 119911 isin 119864 then
100381610038161003816100381610038161199082 minus 119905119908
2
1
10038161003816100381610038161003816le
minus119905 119905 lt minus1
1 minus1 le 119905 le 1
119905 119905 gt 1
(12)
For 119905 le minus1 or 119905 ge 1 the equality holds if and only if 119908(119911) = 119911or one of its rotation For minus1 le 119905 le 1 the equality holds if119908(119911) = 119911
2 or one of its rotation the equality holds for 119905 = minus1if and only if 119908(119911) = 119911((120582 + 119911)(1 + 120582119911)) (0 le 120582 le 1) or oneof its rotation while for 119905 = 1 the equality holds if and only if119908(119911) = minus119911((120582 + 119911)(1 + 120582119911)) (0 le 120582 le 1) or one of its rotationThe above upper bound for minus1 lt 119905 lt 1 is sharp and it can beimproved as follows
100381610038161003816100381610038161199082 minus 119905119908
2
1
10038161003816100381610038161003816+ (1 + 119905)
1003816100381610038161003816119908110038161003816100381610038162le 1 minus1 lt 119905 le 0
100381610038161003816100381610038161199082 minus 119905119908
2
1
10038161003816100381610038161003816+ (1 minus 119905)
1003816100381610038161003816119908110038161003816100381610038162le 1 0 lt 119905 lt 1
(13)
Lemma 4 (see [6 (7) page 10]) If119908 isin Ω and119908(119911) = 1199081119911 +11990821199112+ sdot sdot sdot 119911 isin 119864 then
100381610038161003816100381610038161199082 minus 119905119908
2
1
10038161003816100381610038161003816le max 1 |119905| (14)
for any complex number 119905The result is sharp for the functions119908(119911) = 119911
2 or 119908(119911) = 119911
Lemma 5 (see [7]) If119908 isin Ω then for any real number 1199021 and1199022 the following sharp estimate holds
100381610038161003816100381610038161199083 + 119902111990811199082 + 1199022119908
3
1
10038161003816100381610038161003816le 119867 (1199021 1199022) (15)
where
119867(1199021 1199022)
=
1
(1199021 1199022) isin 1198631 cup 1198632100381610038161003816100381611990221003816100381610038161003816
(1199021 1199022) isin7
⋃119896=3
119863119896
2
3(100381610038161003816100381611990211003816100381610038161003816 + 1) (
100381610038161003816100381611990211003816100381610038161003816 + 1
3 (100381610038161003816100381611990211003816100381610038161003816 + 1 + 1199022)
)
12
(1199021 1199022) isin 1198638 cup 1198639
1
31199022 (
1199022
1minus 4
11990221minus 41199022
)(1199022
1minus 4
3 (1199022 minus 1))
12
(1199021 1199022) isin 11986310 cup 11986311 plusmn2 1
2
3(100381610038161003816100381611990211003816100381610038161003816 minus 1) (
100381610038161003816100381611990211003816100381610038161003816 minus 1
3 (100381610038161003816100381611990211003816100381610038161003816 minus 1 minus 1199022)
)
12
(1199021 1199022) isin 11986312
(16)
Journal of Applied Mathematics 3
The extremal function up to the rotations is of the form
119908 (119911) = 1199113 119908 (119911) = 119911
119908 (119911) = 1199080 (119911) =119911 ([(1 minus 120582) 1205762 + 1205821205761] minus 12057611205762119911)
1 minus [(1 minus 120582) 1205761 + 1205821205762] 119911
119908 (119911) = 1199081 (119911) =119911 (1199051 minus 119911)
1 minus 1199051119911
119908 (119911) = 1199082 (119911) =119911 (1199052 + 119911)
1 + 1199052119911
100381610038161003816100381612057611003816100381610038161003816 =
100381610038161003816100381612057621003816100381610038161003816 = 1 1205761 = 1199050 minus 119890
minus11989412057902 (119886 ∓ 119887)
1205762 = minus119890minus11989412057902 (119894119886 plusmn 119887)
119886 = 1199050 cos1205790
2 119887 = radic1 minus 1199052
0sin2
12057902 120582 =
119887 plusmn 119886
2119887
1199050 = (21199022 (119902
2
1+ 2) minus 3119902
2
1
3 (1199022 minus 1) (11990221minus 41199022)
)
12
1199051 = (
100381610038161003816100381611990211003816100381610038161003816 + 1
3 (100381610038161003816100381611990211003816100381610038161003816 + 1 + 1199022)
)
12
1199052 = (
100381610038161003816100381611990211003816100381610038161003816 minus 1
3 (100381610038161003816100381611990211003816100381610038161003816 minus 1 minus 1199022)
)
12
cos1205790
2=1199021
2(1199022 (1199022
1+ 8) minus 2 (119902
2
1+ 2)
21199022 (11990221+ 2) minus 31199022
1
)
(17)
The sets119863119896 119896 = 1 2 12 are defined as follows
1198631 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 le
1
2100381610038161003816100381611990221003816100381610038161003816 le 1
1198632 = (1199021 1199022) 1
2le100381610038161003816100381611990211003816100381610038161003816 le 2
4
27(100381610038161003816100381611990211003816100381610038161003816 + 1)3minus (100381610038161003816100381611990211003816100381610038161003816 + 1) le 1199022 le 1
1198633 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 le
1
2100381610038161003816100381611990221003816100381610038161003816 le minus1
1198634 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 ge
1
2 1199022 le minus
2
3(100381610038161003816100381611990211003816100381610038161003816 + 1)
1198635 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 le 2 1199022 ge 1
1198636 = (1199021 1199022) 2 le100381610038161003816100381611990211003816100381610038161003816 le 4 1199022 ge
1
12(1199022
1+ 8)
1198637 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 ge 4 1199022 ge
2
3(100381610038161003816100381611990211003816100381610038161003816 minus 1)
1198638 = (1199021 1199022) 1
2le100381610038161003816100381611990211003816100381610038161003816 le 2
minus2
3(100381610038161003816100381611990211003816100381610038161003816 + 1) le 1199022 le
4
27(100381610038161003816100381611990211003816100381610038161003816 + 1)3minus (100381610038161003816100381611990211003816100381610038161003816 + 1)
1198639 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 ge 2
minus2
3(100381610038161003816100381611990211003816100381610038161003816 + 1) le 1199022 le
2100381610038161003816100381611990211003816100381610038161003816 (100381610038161003816100381611990211003816100381610038161003816 + 1)
11990221+ 2
100381610038161003816100381611990211003816100381610038161003816 + 4
11986310 = (1199021 1199022) 2 le100381610038161003816100381611990211003816100381610038161003816 le 4
2100381610038161003816100381611990211003816100381610038161003816 (100381610038161003816100381611990211003816100381610038161003816 + 1)
11990221+ 2
100381610038161003816100381611990211003816100381610038161003816 + 4
le 1199022 le1
12(1199022
1+ 8)
11986311 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 ge 4
2100381610038161003816100381611990211003816100381610038161003816 (100381610038161003816100381611990211003816100381610038161003816 + 1)
11990221+ 2
100381610038161003816100381611990211003816100381610038161003816 + 4
le 1199022 le2100381610038161003816100381611990211003816100381610038161003816 (100381610038161003816100381611990211003816100381610038161003816 minus 1)
11990221minus 2
100381610038161003816100381611990211003816100381610038161003816 + 4
11986312 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 ge 4
2100381610038161003816100381611990211003816100381610038161003816 (100381610038161003816100381611990211003816100381610038161003816 minus 1)
11990221minus 2
100381610038161003816100381611990211003816100381610038161003816 + 4
le 1199022 le2
3(100381610038161003816100381611990211003816100381610038161003816 minus 1)
(18)
2 Main Results
Theorem 6 Let 120601(119911) = 1+1198611119911+11986121199112 +11986131199113 + sdot sdot sdot where 119861119899119904 are real with 1198611 gt 0 and 1198612 ge 0 Let
1205901 =(120572119901 + 120573)
2
11988711990111986121(120572119901 + 2120573)
times 2 (1198612 minus 1198611) minus 1198871199011198612
1
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
1205902 =(120572119901 + 120573)
2
11988711990111986121(120572119901 + 2120573)
times 2 (1198612 + 1198611) minus 1198871199011198612
1
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
1205903 =(120572119901 + 120573)
2
11988711990111986121(120572119901 + 2120573)
times 21198612 minus 1198871199011198612
1
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
Φ (119901 120572 120573 120583) =(120572119901 + 2120573) (2120583 + 120572 minus 1)
2(120572119901 + 120573)2
(19)
4 Journal of Applied Mathematics
If119891(119911) is of the form (1) and belongs to the class119881119901119887120572120573(120601) then
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
le
119887119901
2 (120572119901 + 2120573)(1198612 minus
1198871199011198612
1
2Φ (119901 120572 120573 120583))
120583 lt 1205901
1198871199011198611
2 (120572119901 + 2120573) 1205901 le 120583 le 1205902
minus119887119901
2 (120572119901 + 2120573)(1198612 minus
1198871199011198612
1
2Φ (119901 120572 120573 120583))
120583 gt 1205902
(20)
Furthermore for 1205901 le 120583 le 1205903
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
+1
1198871199011198611(2(1 minus
1198612
1198611)(120572119901 + 120573)
2
(120572119901 + 2120573)
+1198871199011198611
2(2120583 + 120572 minus 1))
10038161003816100381610038161003816119886119901+1
10038161003816100381610038161003816
2
le1198871199011198611
2 (120572119901 + 2120573)
(21)
and for 1205903 le 120583 le 1205902
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
+1
1198871199011198611(2(1 +
1198612
1198611)(120572119901 + 120573)
2
(120572119901 + 2120573)
minus1198871199011198611
2(2120583 + 120572 minus 1))
10038161003816100381610038161003816119886119901+1
10038161003816100381610038161003816
2
le1198871199011198611
2 (120572119901 + 2120573)
(22)
For any complex number 120583
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
le1198871199011198611
2 (120572119901 + 2120573)max1
10038161003816100381610038161003816100381610038161003816
1198871199011198611
2Φ (119901 120572 120573 120583) minus
1198612
1198611
10038161003816100381610038161003816100381610038161003816
(23)
Also
10038161003816100381610038161003816119886119901+3
10038161003816100381610038161003816le
1198871199011198611
2 (120572119901 + 3120573)119867 (1199021 1199022) (24)
where119867(1199021 1199022) is defined in Lemma 3 and
1199021 =21198612
1198611+1198871199011198611
2
(1 minus 120572) (120572119901 + 3120573)
(120572119901 + 120573) (120572119901 + 2120573)
1199022 =1198613
1198611+ (
1198871199011198611
2)
2(120572 minus 1) (2120572 minus 1) (120572119901 + 3120573)
6(120572119901 + 120573)3
+1198871199011198612
2
(1 minus 120572) (120572119901 + 3120573)
(120572119901 + 120573) (120572119901 + 2120573)
(25)
These results are sharp
Proof Since 119891 isin 119881119901119887120572120573(120601) therefore we have for a Schwarzfunction
119908 (119911) = 1199081119911 + 11990821199112+ sdot sdot sdot 119911 isin 119864 (26)
such that
1 minus2
119887+2
119887(1 minus 120573) (
119891(119911)
119911119901)
120572
+ 1205731199111198911015840(119911)
119901119891 (119911)(119891(119911)
119911119901)
120572
= 120601 (119908 (119911))
(27)
Now
1 minus2
119887+2
119887(1 minus 120573) (
119891(119911)
119911119901)
120572
+ 1205731199111198911015840(119911)
119901119891 (119911)(119891(119911)
119911119901)
120572
= 1 +2
119887119901(120572119901 + 120573) 119886119901+1119911 +
1
119887119901(120572119901 + 2120573)
times 2119886119901+2 + (120572 minus 1) 1198862
119901+1 1199112+2 (120572119901 + 3120573)
119887119901
times 119886119901+3 + (120572 minus 1) 119886119901+1119886119901+2
+(120572 minus 1) (120572 minus 2)
61198863
119901+1 1199113+ sdot sdot sdot
(28)
Also we have
120601 (119908 (119911)) = 1 + 11986111199081119911 + (11986111199082 + 11986111199082
1) 1199112
+ (11986111199083 + 2119861211990811199082 + 11986131199083
1) 1199113+ sdot sdot sdot
(29)
Comparing the coefficients of 119911 1199112 1199113 and after simple calcu-lations we obtain
119886119901+1 =11988711990111986111199081
2 (120572119901 + 120573)
119886119901+2 =1198871199011198611
2 (120572119901 + 2120573)
times 1199082 minus (1198871199011198611
2
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
minus1198612
1198611)1199082
1
119886119901+3 =1198871199011198611
2 (120572119901 + 2120573)1199083 + 119902111990811199082 + 1199022119908
3
1
(30)
Journal of Applied Mathematics 5
where 1199021 and 1199022 are defined in (25) It can be easily followedfrom (30) that
119886119901+2 minus 1205831198862
119901+1=
1198871199011198611
2 (120572119901 + 2120573)1199082 minus ]119908
2
1 (31)
where
] =1198871199011198611
2Φ (119901 120572 120573 120583) minus
1198612
1198611 (32)
The results from (20) to (22) are obtained by using Lemma 3(23) by using Lemma 4 and (24) by using Lemma 5 To showthat these results are sharp we define the functions 119870120601119899(119911)119865120582(119911) and 119866120582(119911) such that
119870120601119899 (0) = [119870120601119899]1015840
(0) minus 1 = 0
119865120582 (0) = 1198651015840
120582(0) minus 1 = 0
119866120582 (0) = 1198661015840
120582(0) minus 1 = 0
(33)
with
1 minus2
119887+2
119887(1 minus 120573)(
119870120601119899(119911)
119911119901)
120572
+ 1205731199111198701015840
120601119899(119911)
119901119870120601119899 (119911)(119870120601119899(119911)
119911119901)
120572
≺ 120601 (119911119899minus1)
1 minus2
119887+2
119887(1 minus 120573) (
119865120582(119911)
119911119901)
120572
+ 1205731199111198651015840
120582(119911)
119901119865120582 (119911)(119865120582(119911)
119911119901)
120572
≺ 120601(119911 (119911 + 120582)
1 + 120582119911)
1 minus2
119887+2
119887(1 minus 120573) (
119866120582(119911)
119911119901)
120572
+ 1205731199111198661015840
120582(119911)
119901119866120582 (119911)(119866120582(119911)
119911119901)
120572
≺ 120601(minus119911 (119911 + 120582)
1 + 120582119911)
(34)
It is clear that the functions 119870120601119899 119865120582 119866120582 isin 119881119901119887120572120573(120601) Let119870120601 = 1198701206012 If 120583 lt 1205901 or 120583 gt 1205902 then the equality occursfor the function 119870120601 or one of its rotations For 1205901 lt 120583 lt 1205902the equality is attained if and only if 119891 is 1198701206013 or one ofits rotations When 120583 = 1205901 then the equality holds for thefunction 119865120582 or one of its rotations If 120583 = 1205902 then the equalityis obtained for the function 119866120582 or one of its rotations
Corollary 7 For 119887 = 2 the results from (20) to (24) coincidewith the results proved by Ramachandran et al [8]
Corollary 8 For 119887 = 2 119901 = 1 and 120573 = 1 the results from(20) to (22) coincide with the results obtained by Ravichandranet al [4] for the class 119861120572(120601)
Corollary 9 For 119887 = 2 120572 = 0 and 120573 = 1 the results from(20) to (24) coincide with the results obtained by Ali et al [1]for the class 119878lowast
119901(120601)
Corollary 10 For 119887 = 2 119901 = 1 120572 = 0 and 120573 = 1 the resultsfrom (20) to (22) coincide with the results obtained by Ma andMinda [2] for the class 119878lowast(120601)
21 Application of Theorem 6 tothe Function Defined by Convolutions
Theorem 11 Let 120601(119911) = 1 + 1198611119911 + 11986121199112 + 11986131199113 + sdot sdot sdot where119861119899 119904 are real with 1198611 gt 0 and 1198612 ge 0 Let
1205901 =1198922
119901+1
119892119901+2
(120572119901 + 120573)2
11988711990111986121(120572119901 + 2120573)
times 2 (1198612 minus 1198611)
minus 1198871199011198612
1
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
1205902 =1198922
119901+1
119892119901+2
(120572119901 + 120573)2
11988711990111986121(120572119901 + 2120573)
times 2 (1198612 + 1198611)
minus 1198871199011198612
1
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
1205903 =1198922
119901+1
119892119901+2
(120572119901 + 120573)2
11988711990111986121(120572119901 + 2120573)
times 21198612 minus 1198871199011198612
1
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
Φlowast(119901 120572 120573 120583) =
(120572119901 + 2120573) (2120583 (119892119901+2119892119901+1) + 120572 minus 1)
2(120572119901 + 120573)2
(35)
If 119891(119911) is of the form (1) and belongs to the class 119881119901119887120572120573119892(120601)then
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
le
119887119901
2119892119901+2 (120572119901 + 2120573)(1198612 minus
1198871199011198612
1
2Φlowast(119901 120572 120573 120583))
120583 lt 1205901
1198871199011198611
2119892119901+2 (120572119901 + 2120573) 1205901 le 120583 le 1205902
119887119901
2119892119901+2 (120572119901 + 2120573)(1198871199011198612
1
2Φlowast(119901 120572 120573 120583) minus 1198612)
120583 gt 1205902
(36)
6 Journal of Applied Mathematics
Furthermore for 1205901 le 120583 le 1205903
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
+1198922
119901+1
1198871199011198611119892119901+2(2(1 minus
1198612
1198611)(120572119901 + 120573)
2
(120572119901 + 2120573)
+1198871199011198611
2(2120583
119892119901+2
119892119901+1+ 120572 minus 1))
10038161003816100381610038161003816119886119901+1
10038161003816100381610038161003816
2
le1198871199011198611
2119892119901+2 (120572119901 + 2120573)
(37)
and for 1205903 le 120583 le 1205902
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
+1198922
119901+1
1198871199011198611119892119901+2(2(1 +
1198612
1198611)(120572119901 + 120573)
2
(120572119901 + 2120573)
minus1198871199011198611
2(2120583
119892119901+2
119892119901+1+ 120572 minus 1))
10038161003816100381610038161003816119886119901+1
10038161003816100381610038161003816
2
le1198871199011198611
2119892119901+2 (120572119901 + 2120573)
(38)
For any complex number 120583
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
le1198871199011198611
2119892119901+2 (120572119901 + 2120573)max1
10038161003816100381610038161003816100381610038161003816
1198871199011198611
2Φlowast(119901 120572 120573 120583) minus
1198612
1198611
10038161003816100381610038161003816100381610038161003816
(39)
Also
10038161003816100381610038161003816119886119901+3
10038161003816100381610038161003816le
1198871199011198611
2 (120572119901 + 3120573) 119892119901+3119867(1199021 1199022) (40)
where119867(1199021 1199022) is defined in Lemma 5 and
1199021 =21198612
1198611+1198871199011198611
2
(1 minus 120572) (120572119901 + 3120573)
(120572119901 + 120573) (120572119901 + 2120573)
1199022 =1198613
1198611+ (
1198871199011198611
2)
2(120572 minus 1) (2120572 minus 1) (120572119901 + 3120573)
6(120572119901 + 120573)3
+1198871199011198612
2
(1 minus 120572) (120572119901 + 3120573)
(120572119901 + 120573) (120572119901 + 2120573)
(41)
These results are sharp
Proof Since119891 isin 119881119901119887120572120573119892(120601) therefore we have for a Schwarzfunction 119908 such that
1 minus2
119887+2
119887 (1 minus 120573)(
(119891 lowast 119892) (119911)
119911119901)
120572
+ 120573119911(119891 lowast 119892)
1015840(119911)
119901 (119891 lowast 119892) (119911)((119891 lowast 119892) (119911)
119911119901)
120572
= 120601 (119908 (119911))
(42)
Now
1 minus2
119887+2
119887 (1 minus 120573)(
(119891 lowast 119892) (119911)
119911119901)
120572
+ 120573119911(119891 lowast 119892)
1015840(119911)
119901 (119891 lowast 119892) (119911)((119891 lowast 119892) (119911)
119911119901)
120572
= 1 +2
119887119901(120572119901 + 120573) 119886119901+1119892119901+1119911 +
1
119887119901(120572119901 + 2120573)
times 2119886119901+2119892119901+2 + (120572 minus 1) 1198862
119901+11198922
119901+1 1199112+2
119887119901(120572119901 + 3120573)
times
119886119901+3119892119901+3 + (120572 minus 1) 119886119901+1119886119901+2119892119901+1119892119901+2
+(120572 minus 1) (120572 minus 2)
61198863
119901+11198923
119901+1
1199113+ sdot sdot sdot
(43)
Also we obtain
120601 (119908 (119911)) = 1 + 11986111199081119911 + (11986111199082 + 11986111199082
1) 1199112
+ (11986111199083 + 2119861211990811199082 + 11986131199083
1) 1199113+ sdot sdot sdot
(44)
Comparing the coefficients of 119911 1199112 1199113 and after simplecalculations we obtain
119886119901+1 =11988711990111986111199081
2119892119901+1 (120572119901 + 120573)
119886119901+2 =1198871199011198611
2119892119901+2 (120572119901 + 2120573)
times 1199082 minus (1198871199011198611
2
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
minus1198612
1198611)1199082
1
119886119901+3 =1198871199011198611
2119892119901+3 (120572119901 + 2120573)1199083 + 119902111990811199082 + 1199022119908
3
1
(45)
The remaining proof of the theorem is similar to the proof ofTheorem 6
Corollary 12 For 119887 = 2 the results from (36) to (40) coincidewith the results proved by Ramachandran et al [8] for the class1198771199011120572120573119892(120601)
Journal of Applied Mathematics 7
Corollary 13 For 119887 = 2 120572 = 0 and 120573 = 1 the results from(36) to (40) coincide with the results obtained by Ali et al [1]for the class 119878lowast
119901119892(120601)
Corollary 14 For 119887 = 2 119901 = 1 120572 = 0 and 120573 = 1
1198922 =Γ (3) Γ (2 minus 120582)
Γ (3 minus 120582)=
2
2 minus 120582
1198923 =Γ (4) Γ (2 minus 120582)
Γ (4 minus 120582)=
6
(2 minus 120582) (3 minus 120582)
1198611 =8
1205872 1198612 =
16
31205872
(46)
the results from (36) to (38) coincide with the results obtainedby Srivastava and Mishra [9]
Acknowledgment
The work here is fully supported by LRGSTD2011UKMICT0302
References
[1] R M Ali V Ravichandran and N Seenivasagan ldquoCoefficientbounds for 119901-valent functionsrdquo Applied Mathematics and Com-putation vol 187 no 1 pp 35ndash46 2007
[2] W C Ma and D Minda ldquoA unified treatment of some specialclasses of univalent functionsrdquo in Proceedings of the Conferenceon Complex Analysis pp 157ndash169 International Press 1994
[3] S Owa ldquoProperties of certain integral operatorsrdquo SoutheastAsian Bulletin of Mathematics vol 24 no 3 pp 411ndash419 2000
[4] V Ravichandran A Gangadharan and M Darus ldquoFekete-Szego inequality for certain class of Bazilevic functionsrdquo FarEast Journal of Mathematical Sciences vol 15 no 2 pp 171ndash1802004
[5] M P Chen ldquoOn the regular functions satisfying 119877119890119891(119911)119911 gt120588rdquo Bulletin of the Institute of Mathematics Academia Sinica vol3 no 1 pp 65ndash70 1975
[6] F R Keogh andE PMerkes ldquoA coefficient inequality for certainclasses of analytic functionsrdquo Proceedings of the AmericanMathematical Society vol 20 pp 8ndash12 1969
[7] D V Prokhorov and J Szynal ldquoInverse coefficients for(120572 120573)-convex functionsrdquo Annales Universitatis Mariae Curie-Skłodowska A vol 35 pp 125ndash143 1981
[8] C Ramachandran S Sivasubramanian and H SilvermanldquoCertain coefficient bounds for 119901-valent functionsrdquo Interna-tional Journal of Mathematics and Mathematical Sciences vol2007 Article ID 46576 11 pages 2007
[9] H M Srivastava and A K Mishra ldquoApplications of fractionalcalculus to parabolic starlike and uniformly convex functionsrdquoComputers ampMathematics with Applications vol 39 no 3-4 pp57ndash69 2000
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2 Journal of Applied Mathematics
axis with 120601(0) = 1 and 1206011015840(0) gt 0 A function 119891 isin 119860119901 is in theclass 119881119901119887120572120573(120601) if
1 minus2
119887+2
119887 (1 minus 120573) (
119891 (119911)
119911119901)
120572
+ 1205731199111198911015840(119911)
119901119891 (119911)(119891 (119911)
119911119901)
120572
≺ 120601 (119911)
(7)
where 0 le 120573 le 1 120572 ge 0 and 119887 gt 0
Definition 2 A function 119891 isin 119860119901 is in the class 119881119901119887120572120573119892(120601) if
1 minus2
119887+2
119887 (1 minus 120573)(
(119891 lowast 119892) (119911)
119911119901)
120572
+ 120573119911(119891 lowast 119892)
1015840(119911)
119901 (119891 lowast 119892) (119911)((119891 lowast 119892) (119911)
119911119901)
120572
≺ 120601 (119911)
(8)
where 0 le 120573 le 1 120572 ge 0 and 119887 gt 0 In other words a func-tion 119891 isin 119860119901 is in the class 119881119901119887120572120573119892(120601) if (119891 lowast 119892)(119911) isin
119881119901119887120572120573(120601)We have the following special cases
(i) 119881119901211(120601) coincides with the class 119878lowast119901(120601) introduced
and studied by Ali et al [1](ii) For 119901 = 1 119887 = 2 and 120573 = 1 we have the class 119861120572(120601)
introduced and studied by Ravichandran et al [4](iii) For 119887 = 2 and 120601(119911) = (1 + 119860119911)(1 + 119861119911) the class
1198811199012120572120573(120601) reduces to 119867119901(119860 119861 120572 120573) introduced andstudied by Owa [3]
(iv) For 120601(119911) = (1 + (1 minus 2120574)119911)(1 minus 119911) the class 11988111990121205721(120601)reduces to the class 119861119901(120572 120574) defined as
119861119901 (120572 120574) = 119891 isin 119860119901 Re(1199111198911015840(119911)
119901119891 (119911)(119891 (119911)
119911119901)
120572
) gt 120574
0 le 120574 lt 1 119911 isin 119864
(9)
(v) For 120601(119911) = (1 + (1 minus 2120574)119911)(1 minus 119911) the class 11988111990121205720(120601)is defined as
119891 isin 119860119901 Re(119891 (119911)
119911119901)
120572
gt 120574 0 le 120574 lt 1 119911 isin 119864 (10)
(vi) 1198811201(120601) = 119878lowast(120601) is investigated by Ma and Minda
[2](vii) For 120601(119911) = (1 + (1 minus 2120574)119911)(1 minus 119911) the class 1198811210(120601)
reduces to the class
119861120574 = 119891 isin 1198601 Re119891 (119911)
119911gt 120574 (11)
studied by Chen [5]
We need the following results to obtain our main results
Lemma 3 (see [1]) LetΩ be the class of analytic functions119908normalized by119908(0) = 0 satisfying condition |119908(119911)| lt 1 If119908 isin
Ω and 119908(119911) = 1199081119911 + 11990821199112 + sdot sdot sdot 119911 isin 119864 then
100381610038161003816100381610038161199082 minus 119905119908
2
1
10038161003816100381610038161003816le
minus119905 119905 lt minus1
1 minus1 le 119905 le 1
119905 119905 gt 1
(12)
For 119905 le minus1 or 119905 ge 1 the equality holds if and only if 119908(119911) = 119911or one of its rotation For minus1 le 119905 le 1 the equality holds if119908(119911) = 119911
2 or one of its rotation the equality holds for 119905 = minus1if and only if 119908(119911) = 119911((120582 + 119911)(1 + 120582119911)) (0 le 120582 le 1) or oneof its rotation while for 119905 = 1 the equality holds if and only if119908(119911) = minus119911((120582 + 119911)(1 + 120582119911)) (0 le 120582 le 1) or one of its rotationThe above upper bound for minus1 lt 119905 lt 1 is sharp and it can beimproved as follows
100381610038161003816100381610038161199082 minus 119905119908
2
1
10038161003816100381610038161003816+ (1 + 119905)
1003816100381610038161003816119908110038161003816100381610038162le 1 minus1 lt 119905 le 0
100381610038161003816100381610038161199082 minus 119905119908
2
1
10038161003816100381610038161003816+ (1 minus 119905)
1003816100381610038161003816119908110038161003816100381610038162le 1 0 lt 119905 lt 1
(13)
Lemma 4 (see [6 (7) page 10]) If119908 isin Ω and119908(119911) = 1199081119911 +11990821199112+ sdot sdot sdot 119911 isin 119864 then
100381610038161003816100381610038161199082 minus 119905119908
2
1
10038161003816100381610038161003816le max 1 |119905| (14)
for any complex number 119905The result is sharp for the functions119908(119911) = 119911
2 or 119908(119911) = 119911
Lemma 5 (see [7]) If119908 isin Ω then for any real number 1199021 and1199022 the following sharp estimate holds
100381610038161003816100381610038161199083 + 119902111990811199082 + 1199022119908
3
1
10038161003816100381610038161003816le 119867 (1199021 1199022) (15)
where
119867(1199021 1199022)
=
1
(1199021 1199022) isin 1198631 cup 1198632100381610038161003816100381611990221003816100381610038161003816
(1199021 1199022) isin7
⋃119896=3
119863119896
2
3(100381610038161003816100381611990211003816100381610038161003816 + 1) (
100381610038161003816100381611990211003816100381610038161003816 + 1
3 (100381610038161003816100381611990211003816100381610038161003816 + 1 + 1199022)
)
12
(1199021 1199022) isin 1198638 cup 1198639
1
31199022 (
1199022
1minus 4
11990221minus 41199022
)(1199022
1minus 4
3 (1199022 minus 1))
12
(1199021 1199022) isin 11986310 cup 11986311 plusmn2 1
2
3(100381610038161003816100381611990211003816100381610038161003816 minus 1) (
100381610038161003816100381611990211003816100381610038161003816 minus 1
3 (100381610038161003816100381611990211003816100381610038161003816 minus 1 minus 1199022)
)
12
(1199021 1199022) isin 11986312
(16)
Journal of Applied Mathematics 3
The extremal function up to the rotations is of the form
119908 (119911) = 1199113 119908 (119911) = 119911
119908 (119911) = 1199080 (119911) =119911 ([(1 minus 120582) 1205762 + 1205821205761] minus 12057611205762119911)
1 minus [(1 minus 120582) 1205761 + 1205821205762] 119911
119908 (119911) = 1199081 (119911) =119911 (1199051 minus 119911)
1 minus 1199051119911
119908 (119911) = 1199082 (119911) =119911 (1199052 + 119911)
1 + 1199052119911
100381610038161003816100381612057611003816100381610038161003816 =
100381610038161003816100381612057621003816100381610038161003816 = 1 1205761 = 1199050 minus 119890
minus11989412057902 (119886 ∓ 119887)
1205762 = minus119890minus11989412057902 (119894119886 plusmn 119887)
119886 = 1199050 cos1205790
2 119887 = radic1 minus 1199052
0sin2
12057902 120582 =
119887 plusmn 119886
2119887
1199050 = (21199022 (119902
2
1+ 2) minus 3119902
2
1
3 (1199022 minus 1) (11990221minus 41199022)
)
12
1199051 = (
100381610038161003816100381611990211003816100381610038161003816 + 1
3 (100381610038161003816100381611990211003816100381610038161003816 + 1 + 1199022)
)
12
1199052 = (
100381610038161003816100381611990211003816100381610038161003816 minus 1
3 (100381610038161003816100381611990211003816100381610038161003816 minus 1 minus 1199022)
)
12
cos1205790
2=1199021
2(1199022 (1199022
1+ 8) minus 2 (119902
2
1+ 2)
21199022 (11990221+ 2) minus 31199022
1
)
(17)
The sets119863119896 119896 = 1 2 12 are defined as follows
1198631 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 le
1
2100381610038161003816100381611990221003816100381610038161003816 le 1
1198632 = (1199021 1199022) 1
2le100381610038161003816100381611990211003816100381610038161003816 le 2
4
27(100381610038161003816100381611990211003816100381610038161003816 + 1)3minus (100381610038161003816100381611990211003816100381610038161003816 + 1) le 1199022 le 1
1198633 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 le
1
2100381610038161003816100381611990221003816100381610038161003816 le minus1
1198634 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 ge
1
2 1199022 le minus
2
3(100381610038161003816100381611990211003816100381610038161003816 + 1)
1198635 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 le 2 1199022 ge 1
1198636 = (1199021 1199022) 2 le100381610038161003816100381611990211003816100381610038161003816 le 4 1199022 ge
1
12(1199022
1+ 8)
1198637 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 ge 4 1199022 ge
2
3(100381610038161003816100381611990211003816100381610038161003816 minus 1)
1198638 = (1199021 1199022) 1
2le100381610038161003816100381611990211003816100381610038161003816 le 2
minus2
3(100381610038161003816100381611990211003816100381610038161003816 + 1) le 1199022 le
4
27(100381610038161003816100381611990211003816100381610038161003816 + 1)3minus (100381610038161003816100381611990211003816100381610038161003816 + 1)
1198639 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 ge 2
minus2
3(100381610038161003816100381611990211003816100381610038161003816 + 1) le 1199022 le
2100381610038161003816100381611990211003816100381610038161003816 (100381610038161003816100381611990211003816100381610038161003816 + 1)
11990221+ 2
100381610038161003816100381611990211003816100381610038161003816 + 4
11986310 = (1199021 1199022) 2 le100381610038161003816100381611990211003816100381610038161003816 le 4
2100381610038161003816100381611990211003816100381610038161003816 (100381610038161003816100381611990211003816100381610038161003816 + 1)
11990221+ 2
100381610038161003816100381611990211003816100381610038161003816 + 4
le 1199022 le1
12(1199022
1+ 8)
11986311 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 ge 4
2100381610038161003816100381611990211003816100381610038161003816 (100381610038161003816100381611990211003816100381610038161003816 + 1)
11990221+ 2
100381610038161003816100381611990211003816100381610038161003816 + 4
le 1199022 le2100381610038161003816100381611990211003816100381610038161003816 (100381610038161003816100381611990211003816100381610038161003816 minus 1)
11990221minus 2
100381610038161003816100381611990211003816100381610038161003816 + 4
11986312 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 ge 4
2100381610038161003816100381611990211003816100381610038161003816 (100381610038161003816100381611990211003816100381610038161003816 minus 1)
11990221minus 2
100381610038161003816100381611990211003816100381610038161003816 + 4
le 1199022 le2
3(100381610038161003816100381611990211003816100381610038161003816 minus 1)
(18)
2 Main Results
Theorem 6 Let 120601(119911) = 1+1198611119911+11986121199112 +11986131199113 + sdot sdot sdot where 119861119899119904 are real with 1198611 gt 0 and 1198612 ge 0 Let
1205901 =(120572119901 + 120573)
2
11988711990111986121(120572119901 + 2120573)
times 2 (1198612 minus 1198611) minus 1198871199011198612
1
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
1205902 =(120572119901 + 120573)
2
11988711990111986121(120572119901 + 2120573)
times 2 (1198612 + 1198611) minus 1198871199011198612
1
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
1205903 =(120572119901 + 120573)
2
11988711990111986121(120572119901 + 2120573)
times 21198612 minus 1198871199011198612
1
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
Φ (119901 120572 120573 120583) =(120572119901 + 2120573) (2120583 + 120572 minus 1)
2(120572119901 + 120573)2
(19)
4 Journal of Applied Mathematics
If119891(119911) is of the form (1) and belongs to the class119881119901119887120572120573(120601) then
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
le
119887119901
2 (120572119901 + 2120573)(1198612 minus
1198871199011198612
1
2Φ (119901 120572 120573 120583))
120583 lt 1205901
1198871199011198611
2 (120572119901 + 2120573) 1205901 le 120583 le 1205902
minus119887119901
2 (120572119901 + 2120573)(1198612 minus
1198871199011198612
1
2Φ (119901 120572 120573 120583))
120583 gt 1205902
(20)
Furthermore for 1205901 le 120583 le 1205903
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
+1
1198871199011198611(2(1 minus
1198612
1198611)(120572119901 + 120573)
2
(120572119901 + 2120573)
+1198871199011198611
2(2120583 + 120572 minus 1))
10038161003816100381610038161003816119886119901+1
10038161003816100381610038161003816
2
le1198871199011198611
2 (120572119901 + 2120573)
(21)
and for 1205903 le 120583 le 1205902
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
+1
1198871199011198611(2(1 +
1198612
1198611)(120572119901 + 120573)
2
(120572119901 + 2120573)
minus1198871199011198611
2(2120583 + 120572 minus 1))
10038161003816100381610038161003816119886119901+1
10038161003816100381610038161003816
2
le1198871199011198611
2 (120572119901 + 2120573)
(22)
For any complex number 120583
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
le1198871199011198611
2 (120572119901 + 2120573)max1
10038161003816100381610038161003816100381610038161003816
1198871199011198611
2Φ (119901 120572 120573 120583) minus
1198612
1198611
10038161003816100381610038161003816100381610038161003816
(23)
Also
10038161003816100381610038161003816119886119901+3
10038161003816100381610038161003816le
1198871199011198611
2 (120572119901 + 3120573)119867 (1199021 1199022) (24)
where119867(1199021 1199022) is defined in Lemma 3 and
1199021 =21198612
1198611+1198871199011198611
2
(1 minus 120572) (120572119901 + 3120573)
(120572119901 + 120573) (120572119901 + 2120573)
1199022 =1198613
1198611+ (
1198871199011198611
2)
2(120572 minus 1) (2120572 minus 1) (120572119901 + 3120573)
6(120572119901 + 120573)3
+1198871199011198612
2
(1 minus 120572) (120572119901 + 3120573)
(120572119901 + 120573) (120572119901 + 2120573)
(25)
These results are sharp
Proof Since 119891 isin 119881119901119887120572120573(120601) therefore we have for a Schwarzfunction
119908 (119911) = 1199081119911 + 11990821199112+ sdot sdot sdot 119911 isin 119864 (26)
such that
1 minus2
119887+2
119887(1 minus 120573) (
119891(119911)
119911119901)
120572
+ 1205731199111198911015840(119911)
119901119891 (119911)(119891(119911)
119911119901)
120572
= 120601 (119908 (119911))
(27)
Now
1 minus2
119887+2
119887(1 minus 120573) (
119891(119911)
119911119901)
120572
+ 1205731199111198911015840(119911)
119901119891 (119911)(119891(119911)
119911119901)
120572
= 1 +2
119887119901(120572119901 + 120573) 119886119901+1119911 +
1
119887119901(120572119901 + 2120573)
times 2119886119901+2 + (120572 minus 1) 1198862
119901+1 1199112+2 (120572119901 + 3120573)
119887119901
times 119886119901+3 + (120572 minus 1) 119886119901+1119886119901+2
+(120572 minus 1) (120572 minus 2)
61198863
119901+1 1199113+ sdot sdot sdot
(28)
Also we have
120601 (119908 (119911)) = 1 + 11986111199081119911 + (11986111199082 + 11986111199082
1) 1199112
+ (11986111199083 + 2119861211990811199082 + 11986131199083
1) 1199113+ sdot sdot sdot
(29)
Comparing the coefficients of 119911 1199112 1199113 and after simple calcu-lations we obtain
119886119901+1 =11988711990111986111199081
2 (120572119901 + 120573)
119886119901+2 =1198871199011198611
2 (120572119901 + 2120573)
times 1199082 minus (1198871199011198611
2
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
minus1198612
1198611)1199082
1
119886119901+3 =1198871199011198611
2 (120572119901 + 2120573)1199083 + 119902111990811199082 + 1199022119908
3
1
(30)
Journal of Applied Mathematics 5
where 1199021 and 1199022 are defined in (25) It can be easily followedfrom (30) that
119886119901+2 minus 1205831198862
119901+1=
1198871199011198611
2 (120572119901 + 2120573)1199082 minus ]119908
2
1 (31)
where
] =1198871199011198611
2Φ (119901 120572 120573 120583) minus
1198612
1198611 (32)
The results from (20) to (22) are obtained by using Lemma 3(23) by using Lemma 4 and (24) by using Lemma 5 To showthat these results are sharp we define the functions 119870120601119899(119911)119865120582(119911) and 119866120582(119911) such that
119870120601119899 (0) = [119870120601119899]1015840
(0) minus 1 = 0
119865120582 (0) = 1198651015840
120582(0) minus 1 = 0
119866120582 (0) = 1198661015840
120582(0) minus 1 = 0
(33)
with
1 minus2
119887+2
119887(1 minus 120573)(
119870120601119899(119911)
119911119901)
120572
+ 1205731199111198701015840
120601119899(119911)
119901119870120601119899 (119911)(119870120601119899(119911)
119911119901)
120572
≺ 120601 (119911119899minus1)
1 minus2
119887+2
119887(1 minus 120573) (
119865120582(119911)
119911119901)
120572
+ 1205731199111198651015840
120582(119911)
119901119865120582 (119911)(119865120582(119911)
119911119901)
120572
≺ 120601(119911 (119911 + 120582)
1 + 120582119911)
1 minus2
119887+2
119887(1 minus 120573) (
119866120582(119911)
119911119901)
120572
+ 1205731199111198661015840
120582(119911)
119901119866120582 (119911)(119866120582(119911)
119911119901)
120572
≺ 120601(minus119911 (119911 + 120582)
1 + 120582119911)
(34)
It is clear that the functions 119870120601119899 119865120582 119866120582 isin 119881119901119887120572120573(120601) Let119870120601 = 1198701206012 If 120583 lt 1205901 or 120583 gt 1205902 then the equality occursfor the function 119870120601 or one of its rotations For 1205901 lt 120583 lt 1205902the equality is attained if and only if 119891 is 1198701206013 or one ofits rotations When 120583 = 1205901 then the equality holds for thefunction 119865120582 or one of its rotations If 120583 = 1205902 then the equalityis obtained for the function 119866120582 or one of its rotations
Corollary 7 For 119887 = 2 the results from (20) to (24) coincidewith the results proved by Ramachandran et al [8]
Corollary 8 For 119887 = 2 119901 = 1 and 120573 = 1 the results from(20) to (22) coincide with the results obtained by Ravichandranet al [4] for the class 119861120572(120601)
Corollary 9 For 119887 = 2 120572 = 0 and 120573 = 1 the results from(20) to (24) coincide with the results obtained by Ali et al [1]for the class 119878lowast
119901(120601)
Corollary 10 For 119887 = 2 119901 = 1 120572 = 0 and 120573 = 1 the resultsfrom (20) to (22) coincide with the results obtained by Ma andMinda [2] for the class 119878lowast(120601)
21 Application of Theorem 6 tothe Function Defined by Convolutions
Theorem 11 Let 120601(119911) = 1 + 1198611119911 + 11986121199112 + 11986131199113 + sdot sdot sdot where119861119899 119904 are real with 1198611 gt 0 and 1198612 ge 0 Let
1205901 =1198922
119901+1
119892119901+2
(120572119901 + 120573)2
11988711990111986121(120572119901 + 2120573)
times 2 (1198612 minus 1198611)
minus 1198871199011198612
1
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
1205902 =1198922
119901+1
119892119901+2
(120572119901 + 120573)2
11988711990111986121(120572119901 + 2120573)
times 2 (1198612 + 1198611)
minus 1198871199011198612
1
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
1205903 =1198922
119901+1
119892119901+2
(120572119901 + 120573)2
11988711990111986121(120572119901 + 2120573)
times 21198612 minus 1198871199011198612
1
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
Φlowast(119901 120572 120573 120583) =
(120572119901 + 2120573) (2120583 (119892119901+2119892119901+1) + 120572 minus 1)
2(120572119901 + 120573)2
(35)
If 119891(119911) is of the form (1) and belongs to the class 119881119901119887120572120573119892(120601)then
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
le
119887119901
2119892119901+2 (120572119901 + 2120573)(1198612 minus
1198871199011198612
1
2Φlowast(119901 120572 120573 120583))
120583 lt 1205901
1198871199011198611
2119892119901+2 (120572119901 + 2120573) 1205901 le 120583 le 1205902
119887119901
2119892119901+2 (120572119901 + 2120573)(1198871199011198612
1
2Φlowast(119901 120572 120573 120583) minus 1198612)
120583 gt 1205902
(36)
6 Journal of Applied Mathematics
Furthermore for 1205901 le 120583 le 1205903
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
+1198922
119901+1
1198871199011198611119892119901+2(2(1 minus
1198612
1198611)(120572119901 + 120573)
2
(120572119901 + 2120573)
+1198871199011198611
2(2120583
119892119901+2
119892119901+1+ 120572 minus 1))
10038161003816100381610038161003816119886119901+1
10038161003816100381610038161003816
2
le1198871199011198611
2119892119901+2 (120572119901 + 2120573)
(37)
and for 1205903 le 120583 le 1205902
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
+1198922
119901+1
1198871199011198611119892119901+2(2(1 +
1198612
1198611)(120572119901 + 120573)
2
(120572119901 + 2120573)
minus1198871199011198611
2(2120583
119892119901+2
119892119901+1+ 120572 minus 1))
10038161003816100381610038161003816119886119901+1
10038161003816100381610038161003816
2
le1198871199011198611
2119892119901+2 (120572119901 + 2120573)
(38)
For any complex number 120583
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
le1198871199011198611
2119892119901+2 (120572119901 + 2120573)max1
10038161003816100381610038161003816100381610038161003816
1198871199011198611
2Φlowast(119901 120572 120573 120583) minus
1198612
1198611
10038161003816100381610038161003816100381610038161003816
(39)
Also
10038161003816100381610038161003816119886119901+3
10038161003816100381610038161003816le
1198871199011198611
2 (120572119901 + 3120573) 119892119901+3119867(1199021 1199022) (40)
where119867(1199021 1199022) is defined in Lemma 5 and
1199021 =21198612
1198611+1198871199011198611
2
(1 minus 120572) (120572119901 + 3120573)
(120572119901 + 120573) (120572119901 + 2120573)
1199022 =1198613
1198611+ (
1198871199011198611
2)
2(120572 minus 1) (2120572 minus 1) (120572119901 + 3120573)
6(120572119901 + 120573)3
+1198871199011198612
2
(1 minus 120572) (120572119901 + 3120573)
(120572119901 + 120573) (120572119901 + 2120573)
(41)
These results are sharp
Proof Since119891 isin 119881119901119887120572120573119892(120601) therefore we have for a Schwarzfunction 119908 such that
1 minus2
119887+2
119887 (1 minus 120573)(
(119891 lowast 119892) (119911)
119911119901)
120572
+ 120573119911(119891 lowast 119892)
1015840(119911)
119901 (119891 lowast 119892) (119911)((119891 lowast 119892) (119911)
119911119901)
120572
= 120601 (119908 (119911))
(42)
Now
1 minus2
119887+2
119887 (1 minus 120573)(
(119891 lowast 119892) (119911)
119911119901)
120572
+ 120573119911(119891 lowast 119892)
1015840(119911)
119901 (119891 lowast 119892) (119911)((119891 lowast 119892) (119911)
119911119901)
120572
= 1 +2
119887119901(120572119901 + 120573) 119886119901+1119892119901+1119911 +
1
119887119901(120572119901 + 2120573)
times 2119886119901+2119892119901+2 + (120572 minus 1) 1198862
119901+11198922
119901+1 1199112+2
119887119901(120572119901 + 3120573)
times
119886119901+3119892119901+3 + (120572 minus 1) 119886119901+1119886119901+2119892119901+1119892119901+2
+(120572 minus 1) (120572 minus 2)
61198863
119901+11198923
119901+1
1199113+ sdot sdot sdot
(43)
Also we obtain
120601 (119908 (119911)) = 1 + 11986111199081119911 + (11986111199082 + 11986111199082
1) 1199112
+ (11986111199083 + 2119861211990811199082 + 11986131199083
1) 1199113+ sdot sdot sdot
(44)
Comparing the coefficients of 119911 1199112 1199113 and after simplecalculations we obtain
119886119901+1 =11988711990111986111199081
2119892119901+1 (120572119901 + 120573)
119886119901+2 =1198871199011198611
2119892119901+2 (120572119901 + 2120573)
times 1199082 minus (1198871199011198611
2
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
minus1198612
1198611)1199082
1
119886119901+3 =1198871199011198611
2119892119901+3 (120572119901 + 2120573)1199083 + 119902111990811199082 + 1199022119908
3
1
(45)
The remaining proof of the theorem is similar to the proof ofTheorem 6
Corollary 12 For 119887 = 2 the results from (36) to (40) coincidewith the results proved by Ramachandran et al [8] for the class1198771199011120572120573119892(120601)
Journal of Applied Mathematics 7
Corollary 13 For 119887 = 2 120572 = 0 and 120573 = 1 the results from(36) to (40) coincide with the results obtained by Ali et al [1]for the class 119878lowast
119901119892(120601)
Corollary 14 For 119887 = 2 119901 = 1 120572 = 0 and 120573 = 1
1198922 =Γ (3) Γ (2 minus 120582)
Γ (3 minus 120582)=
2
2 minus 120582
1198923 =Γ (4) Γ (2 minus 120582)
Γ (4 minus 120582)=
6
(2 minus 120582) (3 minus 120582)
1198611 =8
1205872 1198612 =
16
31205872
(46)
the results from (36) to (38) coincide with the results obtainedby Srivastava and Mishra [9]
Acknowledgment
The work here is fully supported by LRGSTD2011UKMICT0302
References
[1] R M Ali V Ravichandran and N Seenivasagan ldquoCoefficientbounds for 119901-valent functionsrdquo Applied Mathematics and Com-putation vol 187 no 1 pp 35ndash46 2007
[2] W C Ma and D Minda ldquoA unified treatment of some specialclasses of univalent functionsrdquo in Proceedings of the Conferenceon Complex Analysis pp 157ndash169 International Press 1994
[3] S Owa ldquoProperties of certain integral operatorsrdquo SoutheastAsian Bulletin of Mathematics vol 24 no 3 pp 411ndash419 2000
[4] V Ravichandran A Gangadharan and M Darus ldquoFekete-Szego inequality for certain class of Bazilevic functionsrdquo FarEast Journal of Mathematical Sciences vol 15 no 2 pp 171ndash1802004
[5] M P Chen ldquoOn the regular functions satisfying 119877119890119891(119911)119911 gt120588rdquo Bulletin of the Institute of Mathematics Academia Sinica vol3 no 1 pp 65ndash70 1975
[6] F R Keogh andE PMerkes ldquoA coefficient inequality for certainclasses of analytic functionsrdquo Proceedings of the AmericanMathematical Society vol 20 pp 8ndash12 1969
[7] D V Prokhorov and J Szynal ldquoInverse coefficients for(120572 120573)-convex functionsrdquo Annales Universitatis Mariae Curie-Skłodowska A vol 35 pp 125ndash143 1981
[8] C Ramachandran S Sivasubramanian and H SilvermanldquoCertain coefficient bounds for 119901-valent functionsrdquo Interna-tional Journal of Mathematics and Mathematical Sciences vol2007 Article ID 46576 11 pages 2007
[9] H M Srivastava and A K Mishra ldquoApplications of fractionalcalculus to parabolic starlike and uniformly convex functionsrdquoComputers ampMathematics with Applications vol 39 no 3-4 pp57ndash69 2000
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Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 3
The extremal function up to the rotations is of the form
119908 (119911) = 1199113 119908 (119911) = 119911
119908 (119911) = 1199080 (119911) =119911 ([(1 minus 120582) 1205762 + 1205821205761] minus 12057611205762119911)
1 minus [(1 minus 120582) 1205761 + 1205821205762] 119911
119908 (119911) = 1199081 (119911) =119911 (1199051 minus 119911)
1 minus 1199051119911
119908 (119911) = 1199082 (119911) =119911 (1199052 + 119911)
1 + 1199052119911
100381610038161003816100381612057611003816100381610038161003816 =
100381610038161003816100381612057621003816100381610038161003816 = 1 1205761 = 1199050 minus 119890
minus11989412057902 (119886 ∓ 119887)
1205762 = minus119890minus11989412057902 (119894119886 plusmn 119887)
119886 = 1199050 cos1205790
2 119887 = radic1 minus 1199052
0sin2
12057902 120582 =
119887 plusmn 119886
2119887
1199050 = (21199022 (119902
2
1+ 2) minus 3119902
2
1
3 (1199022 minus 1) (11990221minus 41199022)
)
12
1199051 = (
100381610038161003816100381611990211003816100381610038161003816 + 1
3 (100381610038161003816100381611990211003816100381610038161003816 + 1 + 1199022)
)
12
1199052 = (
100381610038161003816100381611990211003816100381610038161003816 minus 1
3 (100381610038161003816100381611990211003816100381610038161003816 minus 1 minus 1199022)
)
12
cos1205790
2=1199021
2(1199022 (1199022
1+ 8) minus 2 (119902
2
1+ 2)
21199022 (11990221+ 2) minus 31199022
1
)
(17)
The sets119863119896 119896 = 1 2 12 are defined as follows
1198631 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 le
1
2100381610038161003816100381611990221003816100381610038161003816 le 1
1198632 = (1199021 1199022) 1
2le100381610038161003816100381611990211003816100381610038161003816 le 2
4
27(100381610038161003816100381611990211003816100381610038161003816 + 1)3minus (100381610038161003816100381611990211003816100381610038161003816 + 1) le 1199022 le 1
1198633 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 le
1
2100381610038161003816100381611990221003816100381610038161003816 le minus1
1198634 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 ge
1
2 1199022 le minus
2
3(100381610038161003816100381611990211003816100381610038161003816 + 1)
1198635 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 le 2 1199022 ge 1
1198636 = (1199021 1199022) 2 le100381610038161003816100381611990211003816100381610038161003816 le 4 1199022 ge
1
12(1199022
1+ 8)
1198637 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 ge 4 1199022 ge
2
3(100381610038161003816100381611990211003816100381610038161003816 minus 1)
1198638 = (1199021 1199022) 1
2le100381610038161003816100381611990211003816100381610038161003816 le 2
minus2
3(100381610038161003816100381611990211003816100381610038161003816 + 1) le 1199022 le
4
27(100381610038161003816100381611990211003816100381610038161003816 + 1)3minus (100381610038161003816100381611990211003816100381610038161003816 + 1)
1198639 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 ge 2
minus2
3(100381610038161003816100381611990211003816100381610038161003816 + 1) le 1199022 le
2100381610038161003816100381611990211003816100381610038161003816 (100381610038161003816100381611990211003816100381610038161003816 + 1)
11990221+ 2
100381610038161003816100381611990211003816100381610038161003816 + 4
11986310 = (1199021 1199022) 2 le100381610038161003816100381611990211003816100381610038161003816 le 4
2100381610038161003816100381611990211003816100381610038161003816 (100381610038161003816100381611990211003816100381610038161003816 + 1)
11990221+ 2
100381610038161003816100381611990211003816100381610038161003816 + 4
le 1199022 le1
12(1199022
1+ 8)
11986311 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 ge 4
2100381610038161003816100381611990211003816100381610038161003816 (100381610038161003816100381611990211003816100381610038161003816 + 1)
11990221+ 2
100381610038161003816100381611990211003816100381610038161003816 + 4
le 1199022 le2100381610038161003816100381611990211003816100381610038161003816 (100381610038161003816100381611990211003816100381610038161003816 minus 1)
11990221minus 2
100381610038161003816100381611990211003816100381610038161003816 + 4
11986312 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 ge 4
2100381610038161003816100381611990211003816100381610038161003816 (100381610038161003816100381611990211003816100381610038161003816 minus 1)
11990221minus 2
100381610038161003816100381611990211003816100381610038161003816 + 4
le 1199022 le2
3(100381610038161003816100381611990211003816100381610038161003816 minus 1)
(18)
2 Main Results
Theorem 6 Let 120601(119911) = 1+1198611119911+11986121199112 +11986131199113 + sdot sdot sdot where 119861119899119904 are real with 1198611 gt 0 and 1198612 ge 0 Let
1205901 =(120572119901 + 120573)
2
11988711990111986121(120572119901 + 2120573)
times 2 (1198612 minus 1198611) minus 1198871199011198612
1
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
1205902 =(120572119901 + 120573)
2
11988711990111986121(120572119901 + 2120573)
times 2 (1198612 + 1198611) minus 1198871199011198612
1
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
1205903 =(120572119901 + 120573)
2
11988711990111986121(120572119901 + 2120573)
times 21198612 minus 1198871199011198612
1
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
Φ (119901 120572 120573 120583) =(120572119901 + 2120573) (2120583 + 120572 minus 1)
2(120572119901 + 120573)2
(19)
4 Journal of Applied Mathematics
If119891(119911) is of the form (1) and belongs to the class119881119901119887120572120573(120601) then
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
le
119887119901
2 (120572119901 + 2120573)(1198612 minus
1198871199011198612
1
2Φ (119901 120572 120573 120583))
120583 lt 1205901
1198871199011198611
2 (120572119901 + 2120573) 1205901 le 120583 le 1205902
minus119887119901
2 (120572119901 + 2120573)(1198612 minus
1198871199011198612
1
2Φ (119901 120572 120573 120583))
120583 gt 1205902
(20)
Furthermore for 1205901 le 120583 le 1205903
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
+1
1198871199011198611(2(1 minus
1198612
1198611)(120572119901 + 120573)
2
(120572119901 + 2120573)
+1198871199011198611
2(2120583 + 120572 minus 1))
10038161003816100381610038161003816119886119901+1
10038161003816100381610038161003816
2
le1198871199011198611
2 (120572119901 + 2120573)
(21)
and for 1205903 le 120583 le 1205902
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
+1
1198871199011198611(2(1 +
1198612
1198611)(120572119901 + 120573)
2
(120572119901 + 2120573)
minus1198871199011198611
2(2120583 + 120572 minus 1))
10038161003816100381610038161003816119886119901+1
10038161003816100381610038161003816
2
le1198871199011198611
2 (120572119901 + 2120573)
(22)
For any complex number 120583
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
le1198871199011198611
2 (120572119901 + 2120573)max1
10038161003816100381610038161003816100381610038161003816
1198871199011198611
2Φ (119901 120572 120573 120583) minus
1198612
1198611
10038161003816100381610038161003816100381610038161003816
(23)
Also
10038161003816100381610038161003816119886119901+3
10038161003816100381610038161003816le
1198871199011198611
2 (120572119901 + 3120573)119867 (1199021 1199022) (24)
where119867(1199021 1199022) is defined in Lemma 3 and
1199021 =21198612
1198611+1198871199011198611
2
(1 minus 120572) (120572119901 + 3120573)
(120572119901 + 120573) (120572119901 + 2120573)
1199022 =1198613
1198611+ (
1198871199011198611
2)
2(120572 minus 1) (2120572 minus 1) (120572119901 + 3120573)
6(120572119901 + 120573)3
+1198871199011198612
2
(1 minus 120572) (120572119901 + 3120573)
(120572119901 + 120573) (120572119901 + 2120573)
(25)
These results are sharp
Proof Since 119891 isin 119881119901119887120572120573(120601) therefore we have for a Schwarzfunction
119908 (119911) = 1199081119911 + 11990821199112+ sdot sdot sdot 119911 isin 119864 (26)
such that
1 minus2
119887+2
119887(1 minus 120573) (
119891(119911)
119911119901)
120572
+ 1205731199111198911015840(119911)
119901119891 (119911)(119891(119911)
119911119901)
120572
= 120601 (119908 (119911))
(27)
Now
1 minus2
119887+2
119887(1 minus 120573) (
119891(119911)
119911119901)
120572
+ 1205731199111198911015840(119911)
119901119891 (119911)(119891(119911)
119911119901)
120572
= 1 +2
119887119901(120572119901 + 120573) 119886119901+1119911 +
1
119887119901(120572119901 + 2120573)
times 2119886119901+2 + (120572 minus 1) 1198862
119901+1 1199112+2 (120572119901 + 3120573)
119887119901
times 119886119901+3 + (120572 minus 1) 119886119901+1119886119901+2
+(120572 minus 1) (120572 minus 2)
61198863
119901+1 1199113+ sdot sdot sdot
(28)
Also we have
120601 (119908 (119911)) = 1 + 11986111199081119911 + (11986111199082 + 11986111199082
1) 1199112
+ (11986111199083 + 2119861211990811199082 + 11986131199083
1) 1199113+ sdot sdot sdot
(29)
Comparing the coefficients of 119911 1199112 1199113 and after simple calcu-lations we obtain
119886119901+1 =11988711990111986111199081
2 (120572119901 + 120573)
119886119901+2 =1198871199011198611
2 (120572119901 + 2120573)
times 1199082 minus (1198871199011198611
2
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
minus1198612
1198611)1199082
1
119886119901+3 =1198871199011198611
2 (120572119901 + 2120573)1199083 + 119902111990811199082 + 1199022119908
3
1
(30)
Journal of Applied Mathematics 5
where 1199021 and 1199022 are defined in (25) It can be easily followedfrom (30) that
119886119901+2 minus 1205831198862
119901+1=
1198871199011198611
2 (120572119901 + 2120573)1199082 minus ]119908
2
1 (31)
where
] =1198871199011198611
2Φ (119901 120572 120573 120583) minus
1198612
1198611 (32)
The results from (20) to (22) are obtained by using Lemma 3(23) by using Lemma 4 and (24) by using Lemma 5 To showthat these results are sharp we define the functions 119870120601119899(119911)119865120582(119911) and 119866120582(119911) such that
119870120601119899 (0) = [119870120601119899]1015840
(0) minus 1 = 0
119865120582 (0) = 1198651015840
120582(0) minus 1 = 0
119866120582 (0) = 1198661015840
120582(0) minus 1 = 0
(33)
with
1 minus2
119887+2
119887(1 minus 120573)(
119870120601119899(119911)
119911119901)
120572
+ 1205731199111198701015840
120601119899(119911)
119901119870120601119899 (119911)(119870120601119899(119911)
119911119901)
120572
≺ 120601 (119911119899minus1)
1 minus2
119887+2
119887(1 minus 120573) (
119865120582(119911)
119911119901)
120572
+ 1205731199111198651015840
120582(119911)
119901119865120582 (119911)(119865120582(119911)
119911119901)
120572
≺ 120601(119911 (119911 + 120582)
1 + 120582119911)
1 minus2
119887+2
119887(1 minus 120573) (
119866120582(119911)
119911119901)
120572
+ 1205731199111198661015840
120582(119911)
119901119866120582 (119911)(119866120582(119911)
119911119901)
120572
≺ 120601(minus119911 (119911 + 120582)
1 + 120582119911)
(34)
It is clear that the functions 119870120601119899 119865120582 119866120582 isin 119881119901119887120572120573(120601) Let119870120601 = 1198701206012 If 120583 lt 1205901 or 120583 gt 1205902 then the equality occursfor the function 119870120601 or one of its rotations For 1205901 lt 120583 lt 1205902the equality is attained if and only if 119891 is 1198701206013 or one ofits rotations When 120583 = 1205901 then the equality holds for thefunction 119865120582 or one of its rotations If 120583 = 1205902 then the equalityis obtained for the function 119866120582 or one of its rotations
Corollary 7 For 119887 = 2 the results from (20) to (24) coincidewith the results proved by Ramachandran et al [8]
Corollary 8 For 119887 = 2 119901 = 1 and 120573 = 1 the results from(20) to (22) coincide with the results obtained by Ravichandranet al [4] for the class 119861120572(120601)
Corollary 9 For 119887 = 2 120572 = 0 and 120573 = 1 the results from(20) to (24) coincide with the results obtained by Ali et al [1]for the class 119878lowast
119901(120601)
Corollary 10 For 119887 = 2 119901 = 1 120572 = 0 and 120573 = 1 the resultsfrom (20) to (22) coincide with the results obtained by Ma andMinda [2] for the class 119878lowast(120601)
21 Application of Theorem 6 tothe Function Defined by Convolutions
Theorem 11 Let 120601(119911) = 1 + 1198611119911 + 11986121199112 + 11986131199113 + sdot sdot sdot where119861119899 119904 are real with 1198611 gt 0 and 1198612 ge 0 Let
1205901 =1198922
119901+1
119892119901+2
(120572119901 + 120573)2
11988711990111986121(120572119901 + 2120573)
times 2 (1198612 minus 1198611)
minus 1198871199011198612
1
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
1205902 =1198922
119901+1
119892119901+2
(120572119901 + 120573)2
11988711990111986121(120572119901 + 2120573)
times 2 (1198612 + 1198611)
minus 1198871199011198612
1
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
1205903 =1198922
119901+1
119892119901+2
(120572119901 + 120573)2
11988711990111986121(120572119901 + 2120573)
times 21198612 minus 1198871199011198612
1
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
Φlowast(119901 120572 120573 120583) =
(120572119901 + 2120573) (2120583 (119892119901+2119892119901+1) + 120572 minus 1)
2(120572119901 + 120573)2
(35)
If 119891(119911) is of the form (1) and belongs to the class 119881119901119887120572120573119892(120601)then
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
le
119887119901
2119892119901+2 (120572119901 + 2120573)(1198612 minus
1198871199011198612
1
2Φlowast(119901 120572 120573 120583))
120583 lt 1205901
1198871199011198611
2119892119901+2 (120572119901 + 2120573) 1205901 le 120583 le 1205902
119887119901
2119892119901+2 (120572119901 + 2120573)(1198871199011198612
1
2Φlowast(119901 120572 120573 120583) minus 1198612)
120583 gt 1205902
(36)
6 Journal of Applied Mathematics
Furthermore for 1205901 le 120583 le 1205903
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
+1198922
119901+1
1198871199011198611119892119901+2(2(1 minus
1198612
1198611)(120572119901 + 120573)
2
(120572119901 + 2120573)
+1198871199011198611
2(2120583
119892119901+2
119892119901+1+ 120572 minus 1))
10038161003816100381610038161003816119886119901+1
10038161003816100381610038161003816
2
le1198871199011198611
2119892119901+2 (120572119901 + 2120573)
(37)
and for 1205903 le 120583 le 1205902
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
+1198922
119901+1
1198871199011198611119892119901+2(2(1 +
1198612
1198611)(120572119901 + 120573)
2
(120572119901 + 2120573)
minus1198871199011198611
2(2120583
119892119901+2
119892119901+1+ 120572 minus 1))
10038161003816100381610038161003816119886119901+1
10038161003816100381610038161003816
2
le1198871199011198611
2119892119901+2 (120572119901 + 2120573)
(38)
For any complex number 120583
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
le1198871199011198611
2119892119901+2 (120572119901 + 2120573)max1
10038161003816100381610038161003816100381610038161003816
1198871199011198611
2Φlowast(119901 120572 120573 120583) minus
1198612
1198611
10038161003816100381610038161003816100381610038161003816
(39)
Also
10038161003816100381610038161003816119886119901+3
10038161003816100381610038161003816le
1198871199011198611
2 (120572119901 + 3120573) 119892119901+3119867(1199021 1199022) (40)
where119867(1199021 1199022) is defined in Lemma 5 and
1199021 =21198612
1198611+1198871199011198611
2
(1 minus 120572) (120572119901 + 3120573)
(120572119901 + 120573) (120572119901 + 2120573)
1199022 =1198613
1198611+ (
1198871199011198611
2)
2(120572 minus 1) (2120572 minus 1) (120572119901 + 3120573)
6(120572119901 + 120573)3
+1198871199011198612
2
(1 minus 120572) (120572119901 + 3120573)
(120572119901 + 120573) (120572119901 + 2120573)
(41)
These results are sharp
Proof Since119891 isin 119881119901119887120572120573119892(120601) therefore we have for a Schwarzfunction 119908 such that
1 minus2
119887+2
119887 (1 minus 120573)(
(119891 lowast 119892) (119911)
119911119901)
120572
+ 120573119911(119891 lowast 119892)
1015840(119911)
119901 (119891 lowast 119892) (119911)((119891 lowast 119892) (119911)
119911119901)
120572
= 120601 (119908 (119911))
(42)
Now
1 minus2
119887+2
119887 (1 minus 120573)(
(119891 lowast 119892) (119911)
119911119901)
120572
+ 120573119911(119891 lowast 119892)
1015840(119911)
119901 (119891 lowast 119892) (119911)((119891 lowast 119892) (119911)
119911119901)
120572
= 1 +2
119887119901(120572119901 + 120573) 119886119901+1119892119901+1119911 +
1
119887119901(120572119901 + 2120573)
times 2119886119901+2119892119901+2 + (120572 minus 1) 1198862
119901+11198922
119901+1 1199112+2
119887119901(120572119901 + 3120573)
times
119886119901+3119892119901+3 + (120572 minus 1) 119886119901+1119886119901+2119892119901+1119892119901+2
+(120572 minus 1) (120572 minus 2)
61198863
119901+11198923
119901+1
1199113+ sdot sdot sdot
(43)
Also we obtain
120601 (119908 (119911)) = 1 + 11986111199081119911 + (11986111199082 + 11986111199082
1) 1199112
+ (11986111199083 + 2119861211990811199082 + 11986131199083
1) 1199113+ sdot sdot sdot
(44)
Comparing the coefficients of 119911 1199112 1199113 and after simplecalculations we obtain
119886119901+1 =11988711990111986111199081
2119892119901+1 (120572119901 + 120573)
119886119901+2 =1198871199011198611
2119892119901+2 (120572119901 + 2120573)
times 1199082 minus (1198871199011198611
2
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
minus1198612
1198611)1199082
1
119886119901+3 =1198871199011198611
2119892119901+3 (120572119901 + 2120573)1199083 + 119902111990811199082 + 1199022119908
3
1
(45)
The remaining proof of the theorem is similar to the proof ofTheorem 6
Corollary 12 For 119887 = 2 the results from (36) to (40) coincidewith the results proved by Ramachandran et al [8] for the class1198771199011120572120573119892(120601)
Journal of Applied Mathematics 7
Corollary 13 For 119887 = 2 120572 = 0 and 120573 = 1 the results from(36) to (40) coincide with the results obtained by Ali et al [1]for the class 119878lowast
119901119892(120601)
Corollary 14 For 119887 = 2 119901 = 1 120572 = 0 and 120573 = 1
1198922 =Γ (3) Γ (2 minus 120582)
Γ (3 minus 120582)=
2
2 minus 120582
1198923 =Γ (4) Γ (2 minus 120582)
Γ (4 minus 120582)=
6
(2 minus 120582) (3 minus 120582)
1198611 =8
1205872 1198612 =
16
31205872
(46)
the results from (36) to (38) coincide with the results obtainedby Srivastava and Mishra [9]
Acknowledgment
The work here is fully supported by LRGSTD2011UKMICT0302
References
[1] R M Ali V Ravichandran and N Seenivasagan ldquoCoefficientbounds for 119901-valent functionsrdquo Applied Mathematics and Com-putation vol 187 no 1 pp 35ndash46 2007
[2] W C Ma and D Minda ldquoA unified treatment of some specialclasses of univalent functionsrdquo in Proceedings of the Conferenceon Complex Analysis pp 157ndash169 International Press 1994
[3] S Owa ldquoProperties of certain integral operatorsrdquo SoutheastAsian Bulletin of Mathematics vol 24 no 3 pp 411ndash419 2000
[4] V Ravichandran A Gangadharan and M Darus ldquoFekete-Szego inequality for certain class of Bazilevic functionsrdquo FarEast Journal of Mathematical Sciences vol 15 no 2 pp 171ndash1802004
[5] M P Chen ldquoOn the regular functions satisfying 119877119890119891(119911)119911 gt120588rdquo Bulletin of the Institute of Mathematics Academia Sinica vol3 no 1 pp 65ndash70 1975
[6] F R Keogh andE PMerkes ldquoA coefficient inequality for certainclasses of analytic functionsrdquo Proceedings of the AmericanMathematical Society vol 20 pp 8ndash12 1969
[7] D V Prokhorov and J Szynal ldquoInverse coefficients for(120572 120573)-convex functionsrdquo Annales Universitatis Mariae Curie-Skłodowska A vol 35 pp 125ndash143 1981
[8] C Ramachandran S Sivasubramanian and H SilvermanldquoCertain coefficient bounds for 119901-valent functionsrdquo Interna-tional Journal of Mathematics and Mathematical Sciences vol2007 Article ID 46576 11 pages 2007
[9] H M Srivastava and A K Mishra ldquoApplications of fractionalcalculus to parabolic starlike and uniformly convex functionsrdquoComputers ampMathematics with Applications vol 39 no 3-4 pp57ndash69 2000
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Stochastic AnalysisInternational Journal of
4 Journal of Applied Mathematics
If119891(119911) is of the form (1) and belongs to the class119881119901119887120572120573(120601) then
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
le
119887119901
2 (120572119901 + 2120573)(1198612 minus
1198871199011198612
1
2Φ (119901 120572 120573 120583))
120583 lt 1205901
1198871199011198611
2 (120572119901 + 2120573) 1205901 le 120583 le 1205902
minus119887119901
2 (120572119901 + 2120573)(1198612 minus
1198871199011198612
1
2Φ (119901 120572 120573 120583))
120583 gt 1205902
(20)
Furthermore for 1205901 le 120583 le 1205903
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
+1
1198871199011198611(2(1 minus
1198612
1198611)(120572119901 + 120573)
2
(120572119901 + 2120573)
+1198871199011198611
2(2120583 + 120572 minus 1))
10038161003816100381610038161003816119886119901+1
10038161003816100381610038161003816
2
le1198871199011198611
2 (120572119901 + 2120573)
(21)
and for 1205903 le 120583 le 1205902
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
+1
1198871199011198611(2(1 +
1198612
1198611)(120572119901 + 120573)
2
(120572119901 + 2120573)
minus1198871199011198611
2(2120583 + 120572 minus 1))
10038161003816100381610038161003816119886119901+1
10038161003816100381610038161003816
2
le1198871199011198611
2 (120572119901 + 2120573)
(22)
For any complex number 120583
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
le1198871199011198611
2 (120572119901 + 2120573)max1
10038161003816100381610038161003816100381610038161003816
1198871199011198611
2Φ (119901 120572 120573 120583) minus
1198612
1198611
10038161003816100381610038161003816100381610038161003816
(23)
Also
10038161003816100381610038161003816119886119901+3
10038161003816100381610038161003816le
1198871199011198611
2 (120572119901 + 3120573)119867 (1199021 1199022) (24)
where119867(1199021 1199022) is defined in Lemma 3 and
1199021 =21198612
1198611+1198871199011198611
2
(1 minus 120572) (120572119901 + 3120573)
(120572119901 + 120573) (120572119901 + 2120573)
1199022 =1198613
1198611+ (
1198871199011198611
2)
2(120572 minus 1) (2120572 minus 1) (120572119901 + 3120573)
6(120572119901 + 120573)3
+1198871199011198612
2
(1 minus 120572) (120572119901 + 3120573)
(120572119901 + 120573) (120572119901 + 2120573)
(25)
These results are sharp
Proof Since 119891 isin 119881119901119887120572120573(120601) therefore we have for a Schwarzfunction
119908 (119911) = 1199081119911 + 11990821199112+ sdot sdot sdot 119911 isin 119864 (26)
such that
1 minus2
119887+2
119887(1 minus 120573) (
119891(119911)
119911119901)
120572
+ 1205731199111198911015840(119911)
119901119891 (119911)(119891(119911)
119911119901)
120572
= 120601 (119908 (119911))
(27)
Now
1 minus2
119887+2
119887(1 minus 120573) (
119891(119911)
119911119901)
120572
+ 1205731199111198911015840(119911)
119901119891 (119911)(119891(119911)
119911119901)
120572
= 1 +2
119887119901(120572119901 + 120573) 119886119901+1119911 +
1
119887119901(120572119901 + 2120573)
times 2119886119901+2 + (120572 minus 1) 1198862
119901+1 1199112+2 (120572119901 + 3120573)
119887119901
times 119886119901+3 + (120572 minus 1) 119886119901+1119886119901+2
+(120572 minus 1) (120572 minus 2)
61198863
119901+1 1199113+ sdot sdot sdot
(28)
Also we have
120601 (119908 (119911)) = 1 + 11986111199081119911 + (11986111199082 + 11986111199082
1) 1199112
+ (11986111199083 + 2119861211990811199082 + 11986131199083
1) 1199113+ sdot sdot sdot
(29)
Comparing the coefficients of 119911 1199112 1199113 and after simple calcu-lations we obtain
119886119901+1 =11988711990111986111199081
2 (120572119901 + 120573)
119886119901+2 =1198871199011198611
2 (120572119901 + 2120573)
times 1199082 minus (1198871199011198611
2
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
minus1198612
1198611)1199082
1
119886119901+3 =1198871199011198611
2 (120572119901 + 2120573)1199083 + 119902111990811199082 + 1199022119908
3
1
(30)
Journal of Applied Mathematics 5
where 1199021 and 1199022 are defined in (25) It can be easily followedfrom (30) that
119886119901+2 minus 1205831198862
119901+1=
1198871199011198611
2 (120572119901 + 2120573)1199082 minus ]119908
2
1 (31)
where
] =1198871199011198611
2Φ (119901 120572 120573 120583) minus
1198612
1198611 (32)
The results from (20) to (22) are obtained by using Lemma 3(23) by using Lemma 4 and (24) by using Lemma 5 To showthat these results are sharp we define the functions 119870120601119899(119911)119865120582(119911) and 119866120582(119911) such that
119870120601119899 (0) = [119870120601119899]1015840
(0) minus 1 = 0
119865120582 (0) = 1198651015840
120582(0) minus 1 = 0
119866120582 (0) = 1198661015840
120582(0) minus 1 = 0
(33)
with
1 minus2
119887+2
119887(1 minus 120573)(
119870120601119899(119911)
119911119901)
120572
+ 1205731199111198701015840
120601119899(119911)
119901119870120601119899 (119911)(119870120601119899(119911)
119911119901)
120572
≺ 120601 (119911119899minus1)
1 minus2
119887+2
119887(1 minus 120573) (
119865120582(119911)
119911119901)
120572
+ 1205731199111198651015840
120582(119911)
119901119865120582 (119911)(119865120582(119911)
119911119901)
120572
≺ 120601(119911 (119911 + 120582)
1 + 120582119911)
1 minus2
119887+2
119887(1 minus 120573) (
119866120582(119911)
119911119901)
120572
+ 1205731199111198661015840
120582(119911)
119901119866120582 (119911)(119866120582(119911)
119911119901)
120572
≺ 120601(minus119911 (119911 + 120582)
1 + 120582119911)
(34)
It is clear that the functions 119870120601119899 119865120582 119866120582 isin 119881119901119887120572120573(120601) Let119870120601 = 1198701206012 If 120583 lt 1205901 or 120583 gt 1205902 then the equality occursfor the function 119870120601 or one of its rotations For 1205901 lt 120583 lt 1205902the equality is attained if and only if 119891 is 1198701206013 or one ofits rotations When 120583 = 1205901 then the equality holds for thefunction 119865120582 or one of its rotations If 120583 = 1205902 then the equalityis obtained for the function 119866120582 or one of its rotations
Corollary 7 For 119887 = 2 the results from (20) to (24) coincidewith the results proved by Ramachandran et al [8]
Corollary 8 For 119887 = 2 119901 = 1 and 120573 = 1 the results from(20) to (22) coincide with the results obtained by Ravichandranet al [4] for the class 119861120572(120601)
Corollary 9 For 119887 = 2 120572 = 0 and 120573 = 1 the results from(20) to (24) coincide with the results obtained by Ali et al [1]for the class 119878lowast
119901(120601)
Corollary 10 For 119887 = 2 119901 = 1 120572 = 0 and 120573 = 1 the resultsfrom (20) to (22) coincide with the results obtained by Ma andMinda [2] for the class 119878lowast(120601)
21 Application of Theorem 6 tothe Function Defined by Convolutions
Theorem 11 Let 120601(119911) = 1 + 1198611119911 + 11986121199112 + 11986131199113 + sdot sdot sdot where119861119899 119904 are real with 1198611 gt 0 and 1198612 ge 0 Let
1205901 =1198922
119901+1
119892119901+2
(120572119901 + 120573)2
11988711990111986121(120572119901 + 2120573)
times 2 (1198612 minus 1198611)
minus 1198871199011198612
1
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
1205902 =1198922
119901+1
119892119901+2
(120572119901 + 120573)2
11988711990111986121(120572119901 + 2120573)
times 2 (1198612 + 1198611)
minus 1198871199011198612
1
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
1205903 =1198922
119901+1
119892119901+2
(120572119901 + 120573)2
11988711990111986121(120572119901 + 2120573)
times 21198612 minus 1198871199011198612
1
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
Φlowast(119901 120572 120573 120583) =
(120572119901 + 2120573) (2120583 (119892119901+2119892119901+1) + 120572 minus 1)
2(120572119901 + 120573)2
(35)
If 119891(119911) is of the form (1) and belongs to the class 119881119901119887120572120573119892(120601)then
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
le
119887119901
2119892119901+2 (120572119901 + 2120573)(1198612 minus
1198871199011198612
1
2Φlowast(119901 120572 120573 120583))
120583 lt 1205901
1198871199011198611
2119892119901+2 (120572119901 + 2120573) 1205901 le 120583 le 1205902
119887119901
2119892119901+2 (120572119901 + 2120573)(1198871199011198612
1
2Φlowast(119901 120572 120573 120583) minus 1198612)
120583 gt 1205902
(36)
6 Journal of Applied Mathematics
Furthermore for 1205901 le 120583 le 1205903
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
+1198922
119901+1
1198871199011198611119892119901+2(2(1 minus
1198612
1198611)(120572119901 + 120573)
2
(120572119901 + 2120573)
+1198871199011198611
2(2120583
119892119901+2
119892119901+1+ 120572 minus 1))
10038161003816100381610038161003816119886119901+1
10038161003816100381610038161003816
2
le1198871199011198611
2119892119901+2 (120572119901 + 2120573)
(37)
and for 1205903 le 120583 le 1205902
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
+1198922
119901+1
1198871199011198611119892119901+2(2(1 +
1198612
1198611)(120572119901 + 120573)
2
(120572119901 + 2120573)
minus1198871199011198611
2(2120583
119892119901+2
119892119901+1+ 120572 minus 1))
10038161003816100381610038161003816119886119901+1
10038161003816100381610038161003816
2
le1198871199011198611
2119892119901+2 (120572119901 + 2120573)
(38)
For any complex number 120583
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
le1198871199011198611
2119892119901+2 (120572119901 + 2120573)max1
10038161003816100381610038161003816100381610038161003816
1198871199011198611
2Φlowast(119901 120572 120573 120583) minus
1198612
1198611
10038161003816100381610038161003816100381610038161003816
(39)
Also
10038161003816100381610038161003816119886119901+3
10038161003816100381610038161003816le
1198871199011198611
2 (120572119901 + 3120573) 119892119901+3119867(1199021 1199022) (40)
where119867(1199021 1199022) is defined in Lemma 5 and
1199021 =21198612
1198611+1198871199011198611
2
(1 minus 120572) (120572119901 + 3120573)
(120572119901 + 120573) (120572119901 + 2120573)
1199022 =1198613
1198611+ (
1198871199011198611
2)
2(120572 minus 1) (2120572 minus 1) (120572119901 + 3120573)
6(120572119901 + 120573)3
+1198871199011198612
2
(1 minus 120572) (120572119901 + 3120573)
(120572119901 + 120573) (120572119901 + 2120573)
(41)
These results are sharp
Proof Since119891 isin 119881119901119887120572120573119892(120601) therefore we have for a Schwarzfunction 119908 such that
1 minus2
119887+2
119887 (1 minus 120573)(
(119891 lowast 119892) (119911)
119911119901)
120572
+ 120573119911(119891 lowast 119892)
1015840(119911)
119901 (119891 lowast 119892) (119911)((119891 lowast 119892) (119911)
119911119901)
120572
= 120601 (119908 (119911))
(42)
Now
1 minus2
119887+2
119887 (1 minus 120573)(
(119891 lowast 119892) (119911)
119911119901)
120572
+ 120573119911(119891 lowast 119892)
1015840(119911)
119901 (119891 lowast 119892) (119911)((119891 lowast 119892) (119911)
119911119901)
120572
= 1 +2
119887119901(120572119901 + 120573) 119886119901+1119892119901+1119911 +
1
119887119901(120572119901 + 2120573)
times 2119886119901+2119892119901+2 + (120572 minus 1) 1198862
119901+11198922
119901+1 1199112+2
119887119901(120572119901 + 3120573)
times
119886119901+3119892119901+3 + (120572 minus 1) 119886119901+1119886119901+2119892119901+1119892119901+2
+(120572 minus 1) (120572 minus 2)
61198863
119901+11198923
119901+1
1199113+ sdot sdot sdot
(43)
Also we obtain
120601 (119908 (119911)) = 1 + 11986111199081119911 + (11986111199082 + 11986111199082
1) 1199112
+ (11986111199083 + 2119861211990811199082 + 11986131199083
1) 1199113+ sdot sdot sdot
(44)
Comparing the coefficients of 119911 1199112 1199113 and after simplecalculations we obtain
119886119901+1 =11988711990111986111199081
2119892119901+1 (120572119901 + 120573)
119886119901+2 =1198871199011198611
2119892119901+2 (120572119901 + 2120573)
times 1199082 minus (1198871199011198611
2
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
minus1198612
1198611)1199082
1
119886119901+3 =1198871199011198611
2119892119901+3 (120572119901 + 2120573)1199083 + 119902111990811199082 + 1199022119908
3
1
(45)
The remaining proof of the theorem is similar to the proof ofTheorem 6
Corollary 12 For 119887 = 2 the results from (36) to (40) coincidewith the results proved by Ramachandran et al [8] for the class1198771199011120572120573119892(120601)
Journal of Applied Mathematics 7
Corollary 13 For 119887 = 2 120572 = 0 and 120573 = 1 the results from(36) to (40) coincide with the results obtained by Ali et al [1]for the class 119878lowast
119901119892(120601)
Corollary 14 For 119887 = 2 119901 = 1 120572 = 0 and 120573 = 1
1198922 =Γ (3) Γ (2 minus 120582)
Γ (3 minus 120582)=
2
2 minus 120582
1198923 =Γ (4) Γ (2 minus 120582)
Γ (4 minus 120582)=
6
(2 minus 120582) (3 minus 120582)
1198611 =8
1205872 1198612 =
16
31205872
(46)
the results from (36) to (38) coincide with the results obtainedby Srivastava and Mishra [9]
Acknowledgment
The work here is fully supported by LRGSTD2011UKMICT0302
References
[1] R M Ali V Ravichandran and N Seenivasagan ldquoCoefficientbounds for 119901-valent functionsrdquo Applied Mathematics and Com-putation vol 187 no 1 pp 35ndash46 2007
[2] W C Ma and D Minda ldquoA unified treatment of some specialclasses of univalent functionsrdquo in Proceedings of the Conferenceon Complex Analysis pp 157ndash169 International Press 1994
[3] S Owa ldquoProperties of certain integral operatorsrdquo SoutheastAsian Bulletin of Mathematics vol 24 no 3 pp 411ndash419 2000
[4] V Ravichandran A Gangadharan and M Darus ldquoFekete-Szego inequality for certain class of Bazilevic functionsrdquo FarEast Journal of Mathematical Sciences vol 15 no 2 pp 171ndash1802004
[5] M P Chen ldquoOn the regular functions satisfying 119877119890119891(119911)119911 gt120588rdquo Bulletin of the Institute of Mathematics Academia Sinica vol3 no 1 pp 65ndash70 1975
[6] F R Keogh andE PMerkes ldquoA coefficient inequality for certainclasses of analytic functionsrdquo Proceedings of the AmericanMathematical Society vol 20 pp 8ndash12 1969
[7] D V Prokhorov and J Szynal ldquoInverse coefficients for(120572 120573)-convex functionsrdquo Annales Universitatis Mariae Curie-Skłodowska A vol 35 pp 125ndash143 1981
[8] C Ramachandran S Sivasubramanian and H SilvermanldquoCertain coefficient bounds for 119901-valent functionsrdquo Interna-tional Journal of Mathematics and Mathematical Sciences vol2007 Article ID 46576 11 pages 2007
[9] H M Srivastava and A K Mishra ldquoApplications of fractionalcalculus to parabolic starlike and uniformly convex functionsrdquoComputers ampMathematics with Applications vol 39 no 3-4 pp57ndash69 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 5
where 1199021 and 1199022 are defined in (25) It can be easily followedfrom (30) that
119886119901+2 minus 1205831198862
119901+1=
1198871199011198611
2 (120572119901 + 2120573)1199082 minus ]119908
2
1 (31)
where
] =1198871199011198611
2Φ (119901 120572 120573 120583) minus
1198612
1198611 (32)
The results from (20) to (22) are obtained by using Lemma 3(23) by using Lemma 4 and (24) by using Lemma 5 To showthat these results are sharp we define the functions 119870120601119899(119911)119865120582(119911) and 119866120582(119911) such that
119870120601119899 (0) = [119870120601119899]1015840
(0) minus 1 = 0
119865120582 (0) = 1198651015840
120582(0) minus 1 = 0
119866120582 (0) = 1198661015840
120582(0) minus 1 = 0
(33)
with
1 minus2
119887+2
119887(1 minus 120573)(
119870120601119899(119911)
119911119901)
120572
+ 1205731199111198701015840
120601119899(119911)
119901119870120601119899 (119911)(119870120601119899(119911)
119911119901)
120572
≺ 120601 (119911119899minus1)
1 minus2
119887+2
119887(1 minus 120573) (
119865120582(119911)
119911119901)
120572
+ 1205731199111198651015840
120582(119911)
119901119865120582 (119911)(119865120582(119911)
119911119901)
120572
≺ 120601(119911 (119911 + 120582)
1 + 120582119911)
1 minus2
119887+2
119887(1 minus 120573) (
119866120582(119911)
119911119901)
120572
+ 1205731199111198661015840
120582(119911)
119901119866120582 (119911)(119866120582(119911)
119911119901)
120572
≺ 120601(minus119911 (119911 + 120582)
1 + 120582119911)
(34)
It is clear that the functions 119870120601119899 119865120582 119866120582 isin 119881119901119887120572120573(120601) Let119870120601 = 1198701206012 If 120583 lt 1205901 or 120583 gt 1205902 then the equality occursfor the function 119870120601 or one of its rotations For 1205901 lt 120583 lt 1205902the equality is attained if and only if 119891 is 1198701206013 or one ofits rotations When 120583 = 1205901 then the equality holds for thefunction 119865120582 or one of its rotations If 120583 = 1205902 then the equalityis obtained for the function 119866120582 or one of its rotations
Corollary 7 For 119887 = 2 the results from (20) to (24) coincidewith the results proved by Ramachandran et al [8]
Corollary 8 For 119887 = 2 119901 = 1 and 120573 = 1 the results from(20) to (22) coincide with the results obtained by Ravichandranet al [4] for the class 119861120572(120601)
Corollary 9 For 119887 = 2 120572 = 0 and 120573 = 1 the results from(20) to (24) coincide with the results obtained by Ali et al [1]for the class 119878lowast
119901(120601)
Corollary 10 For 119887 = 2 119901 = 1 120572 = 0 and 120573 = 1 the resultsfrom (20) to (22) coincide with the results obtained by Ma andMinda [2] for the class 119878lowast(120601)
21 Application of Theorem 6 tothe Function Defined by Convolutions
Theorem 11 Let 120601(119911) = 1 + 1198611119911 + 11986121199112 + 11986131199113 + sdot sdot sdot where119861119899 119904 are real with 1198611 gt 0 and 1198612 ge 0 Let
1205901 =1198922
119901+1
119892119901+2
(120572119901 + 120573)2
11988711990111986121(120572119901 + 2120573)
times 2 (1198612 minus 1198611)
minus 1198871199011198612
1
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
1205902 =1198922
119901+1
119892119901+2
(120572119901 + 120573)2
11988711990111986121(120572119901 + 2120573)
times 2 (1198612 + 1198611)
minus 1198871199011198612
1
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
1205903 =1198922
119901+1
119892119901+2
(120572119901 + 120573)2
11988711990111986121(120572119901 + 2120573)
times 21198612 minus 1198871199011198612
1
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
Φlowast(119901 120572 120573 120583) =
(120572119901 + 2120573) (2120583 (119892119901+2119892119901+1) + 120572 minus 1)
2(120572119901 + 120573)2
(35)
If 119891(119911) is of the form (1) and belongs to the class 119881119901119887120572120573119892(120601)then
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
le
119887119901
2119892119901+2 (120572119901 + 2120573)(1198612 minus
1198871199011198612
1
2Φlowast(119901 120572 120573 120583))
120583 lt 1205901
1198871199011198611
2119892119901+2 (120572119901 + 2120573) 1205901 le 120583 le 1205902
119887119901
2119892119901+2 (120572119901 + 2120573)(1198871199011198612
1
2Φlowast(119901 120572 120573 120583) minus 1198612)
120583 gt 1205902
(36)
6 Journal of Applied Mathematics
Furthermore for 1205901 le 120583 le 1205903
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
+1198922
119901+1
1198871199011198611119892119901+2(2(1 minus
1198612
1198611)(120572119901 + 120573)
2
(120572119901 + 2120573)
+1198871199011198611
2(2120583
119892119901+2
119892119901+1+ 120572 minus 1))
10038161003816100381610038161003816119886119901+1
10038161003816100381610038161003816
2
le1198871199011198611
2119892119901+2 (120572119901 + 2120573)
(37)
and for 1205903 le 120583 le 1205902
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
+1198922
119901+1
1198871199011198611119892119901+2(2(1 +
1198612
1198611)(120572119901 + 120573)
2
(120572119901 + 2120573)
minus1198871199011198611
2(2120583
119892119901+2
119892119901+1+ 120572 minus 1))
10038161003816100381610038161003816119886119901+1
10038161003816100381610038161003816
2
le1198871199011198611
2119892119901+2 (120572119901 + 2120573)
(38)
For any complex number 120583
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
le1198871199011198611
2119892119901+2 (120572119901 + 2120573)max1
10038161003816100381610038161003816100381610038161003816
1198871199011198611
2Φlowast(119901 120572 120573 120583) minus
1198612
1198611
10038161003816100381610038161003816100381610038161003816
(39)
Also
10038161003816100381610038161003816119886119901+3
10038161003816100381610038161003816le
1198871199011198611
2 (120572119901 + 3120573) 119892119901+3119867(1199021 1199022) (40)
where119867(1199021 1199022) is defined in Lemma 5 and
1199021 =21198612
1198611+1198871199011198611
2
(1 minus 120572) (120572119901 + 3120573)
(120572119901 + 120573) (120572119901 + 2120573)
1199022 =1198613
1198611+ (
1198871199011198611
2)
2(120572 minus 1) (2120572 minus 1) (120572119901 + 3120573)
6(120572119901 + 120573)3
+1198871199011198612
2
(1 minus 120572) (120572119901 + 3120573)
(120572119901 + 120573) (120572119901 + 2120573)
(41)
These results are sharp
Proof Since119891 isin 119881119901119887120572120573119892(120601) therefore we have for a Schwarzfunction 119908 such that
1 minus2
119887+2
119887 (1 minus 120573)(
(119891 lowast 119892) (119911)
119911119901)
120572
+ 120573119911(119891 lowast 119892)
1015840(119911)
119901 (119891 lowast 119892) (119911)((119891 lowast 119892) (119911)
119911119901)
120572
= 120601 (119908 (119911))
(42)
Now
1 minus2
119887+2
119887 (1 minus 120573)(
(119891 lowast 119892) (119911)
119911119901)
120572
+ 120573119911(119891 lowast 119892)
1015840(119911)
119901 (119891 lowast 119892) (119911)((119891 lowast 119892) (119911)
119911119901)
120572
= 1 +2
119887119901(120572119901 + 120573) 119886119901+1119892119901+1119911 +
1
119887119901(120572119901 + 2120573)
times 2119886119901+2119892119901+2 + (120572 minus 1) 1198862
119901+11198922
119901+1 1199112+2
119887119901(120572119901 + 3120573)
times
119886119901+3119892119901+3 + (120572 minus 1) 119886119901+1119886119901+2119892119901+1119892119901+2
+(120572 minus 1) (120572 minus 2)
61198863
119901+11198923
119901+1
1199113+ sdot sdot sdot
(43)
Also we obtain
120601 (119908 (119911)) = 1 + 11986111199081119911 + (11986111199082 + 11986111199082
1) 1199112
+ (11986111199083 + 2119861211990811199082 + 11986131199083
1) 1199113+ sdot sdot sdot
(44)
Comparing the coefficients of 119911 1199112 1199113 and after simplecalculations we obtain
119886119901+1 =11988711990111986111199081
2119892119901+1 (120572119901 + 120573)
119886119901+2 =1198871199011198611
2119892119901+2 (120572119901 + 2120573)
times 1199082 minus (1198871199011198611
2
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
minus1198612
1198611)1199082
1
119886119901+3 =1198871199011198611
2119892119901+3 (120572119901 + 2120573)1199083 + 119902111990811199082 + 1199022119908
3
1
(45)
The remaining proof of the theorem is similar to the proof ofTheorem 6
Corollary 12 For 119887 = 2 the results from (36) to (40) coincidewith the results proved by Ramachandran et al [8] for the class1198771199011120572120573119892(120601)
Journal of Applied Mathematics 7
Corollary 13 For 119887 = 2 120572 = 0 and 120573 = 1 the results from(36) to (40) coincide with the results obtained by Ali et al [1]for the class 119878lowast
119901119892(120601)
Corollary 14 For 119887 = 2 119901 = 1 120572 = 0 and 120573 = 1
1198922 =Γ (3) Γ (2 minus 120582)
Γ (3 minus 120582)=
2
2 minus 120582
1198923 =Γ (4) Γ (2 minus 120582)
Γ (4 minus 120582)=
6
(2 minus 120582) (3 minus 120582)
1198611 =8
1205872 1198612 =
16
31205872
(46)
the results from (36) to (38) coincide with the results obtainedby Srivastava and Mishra [9]
Acknowledgment
The work here is fully supported by LRGSTD2011UKMICT0302
References
[1] R M Ali V Ravichandran and N Seenivasagan ldquoCoefficientbounds for 119901-valent functionsrdquo Applied Mathematics and Com-putation vol 187 no 1 pp 35ndash46 2007
[2] W C Ma and D Minda ldquoA unified treatment of some specialclasses of univalent functionsrdquo in Proceedings of the Conferenceon Complex Analysis pp 157ndash169 International Press 1994
[3] S Owa ldquoProperties of certain integral operatorsrdquo SoutheastAsian Bulletin of Mathematics vol 24 no 3 pp 411ndash419 2000
[4] V Ravichandran A Gangadharan and M Darus ldquoFekete-Szego inequality for certain class of Bazilevic functionsrdquo FarEast Journal of Mathematical Sciences vol 15 no 2 pp 171ndash1802004
[5] M P Chen ldquoOn the regular functions satisfying 119877119890119891(119911)119911 gt120588rdquo Bulletin of the Institute of Mathematics Academia Sinica vol3 no 1 pp 65ndash70 1975
[6] F R Keogh andE PMerkes ldquoA coefficient inequality for certainclasses of analytic functionsrdquo Proceedings of the AmericanMathematical Society vol 20 pp 8ndash12 1969
[7] D V Prokhorov and J Szynal ldquoInverse coefficients for(120572 120573)-convex functionsrdquo Annales Universitatis Mariae Curie-Skłodowska A vol 35 pp 125ndash143 1981
[8] C Ramachandran S Sivasubramanian and H SilvermanldquoCertain coefficient bounds for 119901-valent functionsrdquo Interna-tional Journal of Mathematics and Mathematical Sciences vol2007 Article ID 46576 11 pages 2007
[9] H M Srivastava and A K Mishra ldquoApplications of fractionalcalculus to parabolic starlike and uniformly convex functionsrdquoComputers ampMathematics with Applications vol 39 no 3-4 pp57ndash69 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Applied Mathematics
Furthermore for 1205901 le 120583 le 1205903
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
+1198922
119901+1
1198871199011198611119892119901+2(2(1 minus
1198612
1198611)(120572119901 + 120573)
2
(120572119901 + 2120573)
+1198871199011198611
2(2120583
119892119901+2
119892119901+1+ 120572 minus 1))
10038161003816100381610038161003816119886119901+1
10038161003816100381610038161003816
2
le1198871199011198611
2119892119901+2 (120572119901 + 2120573)
(37)
and for 1205903 le 120583 le 1205902
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
+1198922
119901+1
1198871199011198611119892119901+2(2(1 +
1198612
1198611)(120572119901 + 120573)
2
(120572119901 + 2120573)
minus1198871199011198611
2(2120583
119892119901+2
119892119901+1+ 120572 minus 1))
10038161003816100381610038161003816119886119901+1
10038161003816100381610038161003816
2
le1198871199011198611
2119892119901+2 (120572119901 + 2120573)
(38)
For any complex number 120583
10038161003816100381610038161003816119886119901+2 minus 120583119886
2
119901+1
10038161003816100381610038161003816
le1198871199011198611
2119892119901+2 (120572119901 + 2120573)max1
10038161003816100381610038161003816100381610038161003816
1198871199011198611
2Φlowast(119901 120572 120573 120583) minus
1198612
1198611
10038161003816100381610038161003816100381610038161003816
(39)
Also
10038161003816100381610038161003816119886119901+3
10038161003816100381610038161003816le
1198871199011198611
2 (120572119901 + 3120573) 119892119901+3119867(1199021 1199022) (40)
where119867(1199021 1199022) is defined in Lemma 5 and
1199021 =21198612
1198611+1198871199011198611
2
(1 minus 120572) (120572119901 + 3120573)
(120572119901 + 120573) (120572119901 + 2120573)
1199022 =1198613
1198611+ (
1198871199011198611
2)
2(120572 minus 1) (2120572 minus 1) (120572119901 + 3120573)
6(120572119901 + 120573)3
+1198871199011198612
2
(1 minus 120572) (120572119901 + 3120573)
(120572119901 + 120573) (120572119901 + 2120573)
(41)
These results are sharp
Proof Since119891 isin 119881119901119887120572120573119892(120601) therefore we have for a Schwarzfunction 119908 such that
1 minus2
119887+2
119887 (1 minus 120573)(
(119891 lowast 119892) (119911)
119911119901)
120572
+ 120573119911(119891 lowast 119892)
1015840(119911)
119901 (119891 lowast 119892) (119911)((119891 lowast 119892) (119911)
119911119901)
120572
= 120601 (119908 (119911))
(42)
Now
1 minus2
119887+2
119887 (1 minus 120573)(
(119891 lowast 119892) (119911)
119911119901)
120572
+ 120573119911(119891 lowast 119892)
1015840(119911)
119901 (119891 lowast 119892) (119911)((119891 lowast 119892) (119911)
119911119901)
120572
= 1 +2
119887119901(120572119901 + 120573) 119886119901+1119892119901+1119911 +
1
119887119901(120572119901 + 2120573)
times 2119886119901+2119892119901+2 + (120572 minus 1) 1198862
119901+11198922
119901+1 1199112+2
119887119901(120572119901 + 3120573)
times
119886119901+3119892119901+3 + (120572 minus 1) 119886119901+1119886119901+2119892119901+1119892119901+2
+(120572 minus 1) (120572 minus 2)
61198863
119901+11198923
119901+1
1199113+ sdot sdot sdot
(43)
Also we obtain
120601 (119908 (119911)) = 1 + 11986111199081119911 + (11986111199082 + 11986111199082
1) 1199112
+ (11986111199083 + 2119861211990811199082 + 11986131199083
1) 1199113+ sdot sdot sdot
(44)
Comparing the coefficients of 119911 1199112 1199113 and after simplecalculations we obtain
119886119901+1 =11988711990111986111199081
2119892119901+1 (120572119901 + 120573)
119886119901+2 =1198871199011198611
2119892119901+2 (120572119901 + 2120573)
times 1199082 minus (1198871199011198611
2
(120572 minus 1) (120572119901 + 2120573)
2(120572119901 + 120573)2
minus1198612
1198611)1199082
1
119886119901+3 =1198871199011198611
2119892119901+3 (120572119901 + 2120573)1199083 + 119902111990811199082 + 1199022119908
3
1
(45)
The remaining proof of the theorem is similar to the proof ofTheorem 6
Corollary 12 For 119887 = 2 the results from (36) to (40) coincidewith the results proved by Ramachandran et al [8] for the class1198771199011120572120573119892(120601)
Journal of Applied Mathematics 7
Corollary 13 For 119887 = 2 120572 = 0 and 120573 = 1 the results from(36) to (40) coincide with the results obtained by Ali et al [1]for the class 119878lowast
119901119892(120601)
Corollary 14 For 119887 = 2 119901 = 1 120572 = 0 and 120573 = 1
1198922 =Γ (3) Γ (2 minus 120582)
Γ (3 minus 120582)=
2
2 minus 120582
1198923 =Γ (4) Γ (2 minus 120582)
Γ (4 minus 120582)=
6
(2 minus 120582) (3 minus 120582)
1198611 =8
1205872 1198612 =
16
31205872
(46)
the results from (36) to (38) coincide with the results obtainedby Srivastava and Mishra [9]
Acknowledgment
The work here is fully supported by LRGSTD2011UKMICT0302
References
[1] R M Ali V Ravichandran and N Seenivasagan ldquoCoefficientbounds for 119901-valent functionsrdquo Applied Mathematics and Com-putation vol 187 no 1 pp 35ndash46 2007
[2] W C Ma and D Minda ldquoA unified treatment of some specialclasses of univalent functionsrdquo in Proceedings of the Conferenceon Complex Analysis pp 157ndash169 International Press 1994
[3] S Owa ldquoProperties of certain integral operatorsrdquo SoutheastAsian Bulletin of Mathematics vol 24 no 3 pp 411ndash419 2000
[4] V Ravichandran A Gangadharan and M Darus ldquoFekete-Szego inequality for certain class of Bazilevic functionsrdquo FarEast Journal of Mathematical Sciences vol 15 no 2 pp 171ndash1802004
[5] M P Chen ldquoOn the regular functions satisfying 119877119890119891(119911)119911 gt120588rdquo Bulletin of the Institute of Mathematics Academia Sinica vol3 no 1 pp 65ndash70 1975
[6] F R Keogh andE PMerkes ldquoA coefficient inequality for certainclasses of analytic functionsrdquo Proceedings of the AmericanMathematical Society vol 20 pp 8ndash12 1969
[7] D V Prokhorov and J Szynal ldquoInverse coefficients for(120572 120573)-convex functionsrdquo Annales Universitatis Mariae Curie-Skłodowska A vol 35 pp 125ndash143 1981
[8] C Ramachandran S Sivasubramanian and H SilvermanldquoCertain coefficient bounds for 119901-valent functionsrdquo Interna-tional Journal of Mathematics and Mathematical Sciences vol2007 Article ID 46576 11 pages 2007
[9] H M Srivastava and A K Mishra ldquoApplications of fractionalcalculus to parabolic starlike and uniformly convex functionsrdquoComputers ampMathematics with Applications vol 39 no 3-4 pp57ndash69 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 7
Corollary 13 For 119887 = 2 120572 = 0 and 120573 = 1 the results from(36) to (40) coincide with the results obtained by Ali et al [1]for the class 119878lowast
119901119892(120601)
Corollary 14 For 119887 = 2 119901 = 1 120572 = 0 and 120573 = 1
1198922 =Γ (3) Γ (2 minus 120582)
Γ (3 minus 120582)=
2
2 minus 120582
1198923 =Γ (4) Γ (2 minus 120582)
Γ (4 minus 120582)=
6
(2 minus 120582) (3 minus 120582)
1198611 =8
1205872 1198612 =
16
31205872
(46)
the results from (36) to (38) coincide with the results obtainedby Srivastava and Mishra [9]
Acknowledgment
The work here is fully supported by LRGSTD2011UKMICT0302
References
[1] R M Ali V Ravichandran and N Seenivasagan ldquoCoefficientbounds for 119901-valent functionsrdquo Applied Mathematics and Com-putation vol 187 no 1 pp 35ndash46 2007
[2] W C Ma and D Minda ldquoA unified treatment of some specialclasses of univalent functionsrdquo in Proceedings of the Conferenceon Complex Analysis pp 157ndash169 International Press 1994
[3] S Owa ldquoProperties of certain integral operatorsrdquo SoutheastAsian Bulletin of Mathematics vol 24 no 3 pp 411ndash419 2000
[4] V Ravichandran A Gangadharan and M Darus ldquoFekete-Szego inequality for certain class of Bazilevic functionsrdquo FarEast Journal of Mathematical Sciences vol 15 no 2 pp 171ndash1802004
[5] M P Chen ldquoOn the regular functions satisfying 119877119890119891(119911)119911 gt120588rdquo Bulletin of the Institute of Mathematics Academia Sinica vol3 no 1 pp 65ndash70 1975
[6] F R Keogh andE PMerkes ldquoA coefficient inequality for certainclasses of analytic functionsrdquo Proceedings of the AmericanMathematical Society vol 20 pp 8ndash12 1969
[7] D V Prokhorov and J Szynal ldquoInverse coefficients for(120572 120573)-convex functionsrdquo Annales Universitatis Mariae Curie-Skłodowska A vol 35 pp 125ndash143 1981
[8] C Ramachandran S Sivasubramanian and H SilvermanldquoCertain coefficient bounds for 119901-valent functionsrdquo Interna-tional Journal of Mathematics and Mathematical Sciences vol2007 Article ID 46576 11 pages 2007
[9] H M Srivastava and A K Mishra ldquoApplications of fractionalcalculus to parabolic starlike and uniformly convex functionsrdquoComputers ampMathematics with Applications vol 39 no 3-4 pp57ndash69 2000
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of