INCLUSION RELATIONS BETWEEN CLASSES OF …downloads.hindawi.com/journals/ijsa/1994/584636.pdf · Jacobi Polynomials, Bessel functions and their generalizations such as hypergeometric

1oumal ofApplied Mathematics and Stochastic Analysis7, Number 1, Spring 1994, 79-89

INCLUSION RELATIONS BETWEEN CLASSESOF HYPERGEOMETRIC FUNCTIONS

O.P. AHUJA

University of Papua New GuineaDepartment of Mathematics

Box 220, University Post OfficePapua, NEW GUINEA

M. JAHANGIRI

California State UniversityDepartment of MathematicsBakersfield, CA 93311 USA

ABSTILCT

The use of hypergeometric functions in univalent function theoryreceived special attention after the surprising application of suchfunctions by de Branges in the proof of the 70-year old BieberbachConjecture. In this paper we consider certain classes of analytic functionsand examine the distortion and containment properties of generalizedhypergeometric hnctions under some operators in these classes.

Key words: Generalized hypergeometr]c functions, analyticfunctions.

AMS (MOS) sub]cct classifications: Primary 33A30; Secon-dary 30C45.

1. INTRODUCTION

The expansions and generating functions involving associated Lagurre,Jacobi Polynomials, Bessel functions and their generalizations such as

hypergeometric functions of one and several variables occur frequently in the

seemingly diverse fields of Physics, Engineering, Statistics, Probability,

Operations Research and other branches of applied mathematics (see e.g. Exton

[7] and Schiff [19]). The use of hypergeometric functions in univalent function

theory received special attention after the surprising application of such functions

by de Branges [6] in the proof of the Bieberbach Conjecture [2]; also see [1].

i’Received" September, 1993. Revised: January, 1994.

2This work was completed while the second author was a Visiting Scholar at the University ofCalifornia, Davis.

Printed in the U.S.A. (C) 1994 by North Atlantic Science Publishing Company 79

80 O.P. AHUJA and M. JAI-IANGII

Merkes-Scott [11], Carlson-$haffer [4] and Ruscheweyh-Singh [181 studied the

starlikeness of certain hypergeometric functions. Miller-Mocanu [12] found a

univalence criterion for such functions. The second author and Silvia [8] andmore recently Noor [14] studied the behavior of certain hypergeometric functions

under various operators. In the present paper we consider certain classes of

analytic functions and examine the distortion and containment properties of

generalized hypergeometric functions under some operators in these classes.

Let p and q be natural numbers such that p<_q+l. For aj

(j = 1,2,...,p) and (k = 1,2,...,q) complex numbers such tha Z 0,

-1,- 2,... (k = 1, 2,..., q), let ,Fq(z), that is

()()-..() zApFq(tl’a2"’"aP;/l’fl2"’"flq;Z)’-n=OE (fll)n(fl2)n" "(#q)n nl. (1.1)

denote the generalized hypergeomeric function.

symbol defined by

D(,k + n) [ i,(A),=..- F(X)’ [ A(A+Z)...(A+n-Z),

Here (A). is the Pochhammer

if n-0

if n 1,2,

It is known [9, p. 43] that the series given by (1.1) converges absolutely for

]z[ <oc if p<q+l, and for zU-{z:[z] <1} if p=q+l. Thus for

p<_q+l,

F(,, .,; 5,, 5, ., &; z) ,where .A denotes the class of functions hat are analytic in U.

For oFq(z) defined by (1.1), le

()._ ()._ ...(%)_ (1.2)

for all z U, where p _< q + 1.

Let S be the family of functions of the form

f(z) = z -t-. E anZ (z e u) (.a)

that are analytic and univalent in U.A by

For f S, define the convolution operator

Inclusion Relations Between Classes of Hypergeometric Functions 81

A(Cel, o2,..., Op; ill, f12,’" ", flq; f): -- paqf.

The operator "," denotes the Hadmard product or convolution of two power

series

that is

f(z) = a,.,z’ and g(z)= b,.,z’n=0 n=0

(f*g)(z) = a,.,b,zr’.

Thus for f of the form (1.3) in S, we may write

oo (Ol)n- I((R2)n -1"" "((Y’p)n-1 anzn(z e U),

where p _< q + 1.

We recall that a function is convex in U if it is univalent conformal

mapping of U onto a convex domain. It is well-known that I) is convex in U if

and only if

+ > o (z e u)

and ’ O. Also, a function f is said to be close-to-convex in U if there exists

convex function (I) in U such that Re(f’(z)/ff2’(z)) > 0 (z E U). For 0 a < 1, we

define the following subclasses of A:

A(p, q; a): { ,Fq e A: Re ,Fq(z) >

R(a, r): = {f e A" Ref’(z) > a for

S(a): = {f A: Re(f(z)/z)>

T: = {f e S: f(z) = z-- akzk z e U, ak > O},

k=2

T*(a)" = {f e T: Re(zf’(z)/f(z)) >

C(a): = {f e T: Re(1 + zf"(z)/f’(z)) >

82 O.P. AHUJA and M. JArIANGI

then

2. MAIN RESULTS

We use the following lemma due to Chen [5] to prove our first theorem.

Lemma 1: If f S(a), then

Ref’(z) >_a 2at2 + (2a- 1)r4

(i ,’)

if O <_r < l/2,

if l/2 < r < l.

Theorem 1: Let

(i)

(ii)

ForO_< [zl -r<l/2, z-+Gq+ 6 A(p+l,q+l;X), where

For1 zl -r < l, z-X+Gq+x A(P+l,q+l;Y), where

(1 O)r4Y = 3’(1 r)’ 7 >- 1.

Proof: We observe that

(i i/7) ,G(z) +z ,G’(z) : +G + (z).

Thus by an application of Lemma 1, we have

Re(z - + 1Gq +i(z)) = (1 1/7)ie(z -’ Gv(z)) + }Re( G’q(z))

>_ (I- I/7)a di + 2(2a- 1)r+ (2a- l)r

7(i + )

=X, if0_< Izl =r<l/2,

anda 2ar + (2a 1)r4. + Gq + (z)) >_ (1 1/7)a + 7( 1 r2)


=Y, ill/2_< Izl =r<l.

To prove our next theorem we need the following lemma due to

MacGregor [10].

Lemma 2:

f,(z) < + z

and

z + 2oa(Z + z I) < If(z)! < z 2tog(X z I).

Theorem 2: Let

= "[ Jt-lFq(t) dt-r(prq(z)) ’z ’-i0

/f .r(,Fq) e R(0,1), then

(i) pFq(z)[< 1 +-(i i’ll’ + log(1 z l,

(ill F(z)[ >_ 1 ---(1 -llz[ -" 1--’7=1z1= foe(1 it- Z I))"

Proof: Note that

(7>_ I).

(I- 1/"/).r(vFq(z))-F z(q(,Fq(z))= z ,Fq(z).

Since

Re(q’.r(vFq(z)) > O,

we obtain from (2.1) and Lemma 2

Fq()I_< z

( 1 1-"/log(l_lz )),=-1+- i lki + -and

Fq(z)]>_ (I 1/-r) z

>-(1-1/7)+2(1-1/7) log(l + z )-(l + z )

84 O.P. AHUJA and M. JAHANGI

Theorem 3: If oFg A(p, q; a), 0 <_ a < 1, then for 7 >- 1,

Proof: We observe that

+ Fq + (c, a,..., at, 3’; 3x, fl,..., flq, 3’ + 1; z) = z7_ i ft- f(t)dt.0

Now by applying a result of Own [15] that if f S(a), then

we obtain

f(z)Z < 1 +(1--2 )1zl

1-lzl

3’t _7)dt

0

7 1 1+(I --20)Izl

/((1-I- (1- 2) z I)

Next we use a result of Nikolaeva and Repnina [13] concerning the convex

combination of certain analytic functions to prove our Theorem 4.

Lemma 3: Let f e R(O, 1). Denote

Let

#3h0= 0_<#_<1.(1 + v/i

Ft,(z) (1 #)f(z) + l,zf’(z), h c/(1 c).

Then F. e R(a, r) where

r

ifh>_ho

ifO<h<ho.

Inclusion Relations Between Classes ofHypergeometc Functions 85

Theorem 4: If Gq R(a, 1), 0 < a < 1, then the function

+Gq+x(z) = z+Fq+x(a,a,...,a,l +7; x,,...,#,7;z)is in R(a, r) where

(it) r = r(,7) =

g 0 g. < /( +( +(+ a))) fo .Proof: Noting that

wigh Lemma 3, the resulg follows by simpleand comparing

manipulations.

Corollary: If Gq R(c, 1),univalence of p + iGq + for 7 >- 1.

Theorem 5" If

algebraic

then r = r(a,7) is also the radius of

is convex, close-to-convex, or starlike of order a(O _< a < 1), then so is

z r,+Fq+(al,a2,...,ap,7 + 1;/,52,...,flq,7 + 2;z),

for Re(7) > 0.

Then

Proof: Consider the general transform

7(f(z)) 7z,+ 1 f tT_ f(t)dt, (Re(7) > 0).0

.,(z .F(z)) +2"7 ’(z pFq(t))dt

zn+ln!

86 O.P. AHUJA and M. JAIGI1%I

where

"r+ lz,H() =

is known [16] to be convex in U. It follows from the work of Ruscheweyh andSheil-$mall [17] that the function (I).r(z Fg) is convex, close-o-convex, or sarlike

of order a whenever f is such. Now he result follows because we may write

.r(z vFq(z))= z(a)"(a)""’(av)"(’7 + 1),, z..y.

. =,o (/),,()...(),( + ), !

Our last theorem deals wih hypergeomeric functions with negativecoefficients. Its proof uses the following lemma which is due to Silverman [20].

Lemma 4: Let

(i)

f(z)=z- Eaz, a>O.

f e T*(a). (n a)a. <_ 1 a,

(ii) f e C(a). E n(n a)a. <_ l a,n=2

(iii) f e T*(a)=[a. _<’-",=. (n _> 2),(iv) f e C(a)= a, <_ ,(-%) (n >_ 2).

Theorem 6: For"

f(z) z- a,zk e T*(a),

the function


P

E(-) > ,+,./=1

Proof: For

and using (1.5), we have

where

f(z) = z-- akzk e T

(Ol)k- l(O2)k- 1)" .(Op + 1)k-1ak>O.d, (/l)k- l(2)k- 1’" "(p)k-l(1)k 1

In view of Lemma 4, the function given by (2.3) is in T*(a)if and only if

1--o:

_( (a)!a):.,(a+ !)t; 1-_o ()(&)...(G) ]

= ,+F,(a,a2,...,a,+;fl,fl2,...,fl,;1)- 1.

The desired result follows because the

Izl = l if+F series is absolutely convergent for

see [9, p. 44].

(( )- ,, + ,) > o;k=l

Remark 1" Using (ii) and (iv)in Lemma 4, we can similarly prove

that Theorem 6 holds for C(a), that is,

88 O.P. AHUJA and M. JAHANGI

if and only if the conditions of Theorem 6 hold.

Remark 2: In view of the generalization of the Gaussian summation

formula for p = 2, 3,..., determined in [3], the condition (2.2) may be expressed&S

(s) Ao) < sr()...r(, + )r( + )r( + )(;=s)(+ S) r(z,)r(z;)...r(z)r()

where AP)’s are given by some lengthy expressions in [3] andp

= E (/- )-+ > 0.

REFEIENCES

[1]

[2]

Ahuja, O.P., the Bieberbach Conjecture and its impact on the development in geometricfunction theorem, Mathematical Chronicle 15 (1986), 1-28.

Bieberbach, L., ber die Koeffizienten derjenigen Potenzreihen, welche eine schlichteAbbildung des Einheitskreises vermitten, S.-B. Preuss, Wiss. (1916), 940-955.

[3] Bhring, W., Generalized hypergeometric functions at unit argument, Proc. Amer.Math. 114:1 (1992), 145-153.

[4] Carlson, B.C., and Shaffer, D.B., Starlike and prestarlike hypergeometric functions,SIAM J. Math. Anal. 15 (1984), 737-745.

Chen, Ming-Po, On the regular functions satisfying Re(f(z)/z)> a, Bull. Inst. Math.Acad. Sinica 3:1 (1975), 65-70.

[] de Branges, L., A proof of the Bieberbach Conjecture, Preprint E-5-84, Sleklov Math.Institute, Lomi, Leningrad (1984), 1-12 or Acta Mathematica 154 (1985), 137-152.

[7] Exton, H., Multiple Hypergeometric Functions and Application, John Wiley and Sons(Halsted Press), New York; Ellis Harwood, Chichester 1976.

Is] Jahangiri, M., and Silvia, E.M., Some inequalities involving generalized hypergeometricfunctions, in" Univalent Functions, Fractional Calculus and their Applications, JohnWiley and Sons (Halsted Press), New York; Ellis Harwood, Chichester 1989, 65-74.

[9] Luke, Y.L., The Special Funclions and lheir Approzimations, Vol. I, Academic Press,New York 1969.

[0] MacGregor, T.H., Functions whose derivative has a positive real part, Trans. Amer.Math. Soc. 104 (1962), 532-537.


[II]

[i]

[13]

[14]

[5]

[16]

[17]

[18]

[19]

[20]

Merkes, E.P., and Scott, W.T., Starlike functions, Proc. Amer. Math. Soc. 12 (1961),885-888.

Miller, S.S., and Mocanu, P.T., Differential subordinations and inequalities in thecomplex plane, j. Diff. Equations 67 (1987), 199-211.

Nikolaeva, I.V., and Repnina, L.G., A generalization of a theorem of Livingston,Ukrainskii Mate. Zhurral 24:2 (1972), 268-273.

Noor, K.I., Some results on certain subclasses of analytic functions involving generalizedhypergeometric functions and Hadamard product, Iniernat. J. Math. Math. Sci. 15(1992), 143-148.

Owa, S., On functions satisfying Re(f(z)/z) > o, Tamkang J. Maih. 16:3 (1985), 35-44.

l:tuscheweyh, S., New criteria for univalent functions, Proc. Amer. Math. Soc. 49 (1975),109-115.

luscheweyh, S., and Sheil-Small, T., Hadamard product of schlicht functions andPolya-Schoenberg conjecture, Comment. Math. Helv. 48 (1973), 119-135.

Puscheweyh, S., and Singh, V., On the order of starlikeness of hypergeometric functions,J. Math. Anal. Appl. 113 (1986), 1-11.

Schiff, L., Quantum Mechanics, McGraw-Hill, New York 1955.

Silverman, H., Univalent functions with negative coefficients, Proc. A mer. Math. Soc.51:1 (1975), 109-116.

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INCLUSION RELATIONS BETWEEN CLASSES OF …downloads.hindawi.com/journals/ijsa/1994/584636.pdf · Jacobi Polynomials, Bessel functions and their generalizations such as hypergeometric