# Univalence of a New General Integral Operator Associated with theq-Hypergeometric Function

Authors: Huda Aldweby; Maslina Darus
Journal: International Journal of Mathematics and Mathematical Sciences
 (2013)

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## Abstract

Motivated by the familiarq-hypergeometric functions, we introduce a new family of integral operators and obtain new sufficient conditions of univalence criteria. Several corollaries and consequences of the main results are also pointed out.


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## Body

## 1. Introduction

Let𝒜 denote the class of functions of the form
(1)f(z)=z+∑n=2∞‍cnzn,cn≥0,
which are analytic in the open unit disk 𝒰={z∈ℂ:|z|<1}, and 𝒮 the class of functions f∈𝒜 which are univalent in 𝒰.Letf,g∈𝒜, where f is defined by (1) and g is given by
(2)g(z)=z+∑n=2∞‍bnzn,bn≥0.
Then the Hadamard product (or convolution) f*g of the functions f and g is defined by
(3)(f*g)(z)=z+∑n=2∞‍cnbnzn.For complex parametersai,bj,andq(i=1,…,r,j=1,…,s,bj∈ℂ∖{0,-1,-2,…},|q|<1), we define the q-hypergeometric function rΦs(a1,…,ar;b1,…,bs;q,z) by
(4)rΦs(ai;bj;q,z)=∑n=0∞‍(a1,q)n⋯(ar,q)n(q,q)n(b1,q)n⋯(bs,q)nzn(r=s+1;r,s∈ℕ0=ℕ∪{0};z∈𝒰), where ℕ denotes the set of positive integers and (a,q)n is the q-shifted factorial defined by
(5)(a,q)n={1,n=0;(1-a)(1-aq)(1-aq2)⋯(1-aqn-1),n∈ℕ.By using the ratio test, we should note that, if|q|<1, the series (4) converges absolutely for |z|<1 if r=s+1. For more mathematical background of these functions, one may refer to [1].Corresponding to the function defined by (4), consider (6)r𝒢s(ai;bj;q,z)=zrΦs(ai;bj;q,z).Recently, the authors [2] defined the linear operator ℳ(ai,bj;q)f:𝒜→𝒜 by
(7)ℳ(ai,bj;q)f(z)=r𝒢s(ai;bj;q,z)*f(z)=z+∑n=2∞‍Υncnzn,
where
(8)Υn=(a1,q)n-1⋯(ar,q)n-1(q,q)n-1(b1,q)n-1⋯(bs,q)n-1,(|q|<1).It should be remarked that the linear operator (7) is a generalization of many operators considered earlier. For ai=qαi,bj=qβj,αi,βj∈ℂ,βj≠0,-1,-2,…,(i=1,…,r,j=1,…,s), and q→1, we obtain the Dziok-Srivastava linear operator [3] (forr=s+1), so that it includes (as its special cases) various other linear operators introduced and studied by Ruscheweyh [4], Carlson and Shaffer [5] and the Bernardi-Libera-Livingston operators [6–8].Theq-difference operator is defined by
(9)dqh(z)=h(qz)-h(z)(q-1)z,q≠1,z≠0,limq→1dqh(z)=h′(z),
where h′(z) is the ordinary derivative. For more properties of dq see [9, 10].Lemma 1 (see [2]).
Letf∈𝒜; then (i)
forr=1,s=0, and a1=q, one has ℳ(q,-;q)f(z)=f(z).(ii)
Forr=1,s=0, and a1=q2, one has ℳ(q2,-;q)f(z)=zdqf(z) and limq→1ℳ(q2,-;q)f(z)=zf′(z), where dq is the q-derivative defined by (9).Definition 2.
A functionf∈𝒜 is said to be in the class 𝔅sr(ai,bj;q;μ) if it is satisfying the condition
(10)|z2(ℳ(ai,bj;q)f(z))′[ℳ(ai,bj;q)f(z)]2-1|<1-μ(z∈𝒰;0≤μ<1),
where ℳ(ai,bj;q)f is the operator defined by (7).Note that𝔅01(q,-;q;μ)=𝔅(μ), where the class 𝔅(μ) of analytic and univalent functions was introduced and studied by Frasin and Darus [11].Using the operatorℳ(ai,bj;q)f(z)f, we now introduce the following new general integral operator.Form∈ℕ∪{0}, γ1,γ2,…,γm,δ∈ℂ∖{0,-1,-2,…}, and |q|<1, we define the integral operator Iγk,δ(ai,bj;q;z):𝒜n→𝒜n by
(11)Iγk,δ(ai,bj;q;z)=(δ∫0z‍tδ-1∏k=1m‍(ℳ(ai,bj;q)f(z)fk(t)t)1/γkdt)1/δ,
where fk∈𝒜.Remark 3.
It is interesting to note that the integral operatorIγk,δ(ai,bj;q;z) generalizes many operators introduced and studied by several authors, for example,(1) forr=s+1,ai=qαi,bj=qβj,i=1,…,r,j=1,…,s,q→1,γk=1/(α-1), and δ=1+m(α-1), where α∈ℂ and ℜ(α)>0, we obtain the following integral operator introduced and studied by Selvaraj and Karthikeyan [12]:
(12)Fα(α1,β1;z)=({∫0z‍}1+m(α-1)∫0z‍(Hsr(α1,β1)f1(t))α-1⋯(Hsr(α1,β1)fm(t))α-1dt{∫0z‍})1/(1+m(α-1)),
where for convenience Hsr(α1,β1)f:=H(α1,…,αr;β1,…,βs;z)f(z), and Hsr(α1,β1)f(z)=z+∑n=2∞((α1)n-1⋯(αr)n-1/(β1)n-1⋯(βs)n-1(n-1)!)‍anzn is the Dziok-Srivastava operator [3].(2) Forr=1,s=0,a1=q,γk=1/(α-1), and δ=1+m(α-1), we obtain the integral operator
(13)Fm,α(z)=({∫0z‍(f1(t))α-1}1+m(α-1)×∫0z‍(f1(t))α-1⋯(fm(t))α-1dt{∫0z‍})1/(1+m(α-1))
studied recently by Breaz et al. [13].(3) Forr=1,s=0,a1=q,γk=1/αk, and δ=1, we obtain the integral operator
(14)Fα(z)=∫0z‍(f1(t)t)α1⋯(fm(t)t)αmdt
introduced and studied by D. Breaz and N. Breaz [14].(4) Forr=1,s=0,a1=q2,γk=1/(α-1), and δ=1+m(α-1), we obtain the integral operator
(15)Gα(z)=({∫0z‍tm(α-1)}1+m(α-1)×∫0z‍tm(α-1)(f1′(t))α-1⋯(fm′(t))α-1dt{∫0z‍})1/(1+m(α-1))
introduced by Selvaraj and Karthikeyan [12].(5) Forr=1,s=0,a1=q2,γk=1/α, and δ=1, we obtain the integral operator
(16)Gα(z)=∫0z‍(f1′(t))α⋯(fm′(t))αdt,
recently introduced and studied by Breaz and Güney [15].(6) Forr=1,s=0,a1=q,f1=⋯=fm=f∈𝒜,γk=1/(α-1), and δ=α, where α∈ℂ and ℜ(α)>0, we obtain the integral operator
(17)Gα(z)=(α∫0z‍(f(t))α-1)1/αdt,
introduced and studied by Pescar [16].In order to derive our main results, we have to recall the following univalence criteria.Lemma 4 (see [17, 18]).
Letδ∈ℂ with Re(δ)>0. If f∈𝒜 satisfies
(18)1-|z|2Re(δ)Re(δ)|zf′′(z)f′(z)|≤1,
for all z∈𝒰, then the integral operator
(19)Fδ(z)={δ∫0z‍tδ-1f′(t)dt}1/δ
is in the class 𝒮.Lemma 5 (see [16]).
Letδ∈ℂ with Re(δ)>0,c∈ℂ, with |c|≤1,c≠-1. If f∈𝒜 satisfies
(20)|c|z|2δ+(1-|z|2δ)zf′′(z)δf′(z)|≤1,
for all z∈𝒰 then the integral operator
(21)Fδ(z)={δ∫0z‍tδ-1f′(t)dt}1/δ
is in the class 𝒮.Lemma 6 (Generalized Schwarz Lemma, see [19]).
(Generalized Schwarz Lemma) Let the functionf be analytic in the disk 𝒰R={z:|z|<R}, with |f(z)|<M for fixed M. If f(z) has one zero with multiplicity order bigger that m for z=0, then
(22)|f(z)|≤MRm|z|m,(z∈𝒰R).
Equality can hold only if
(23)f(z)=eiθ(MRm)zm,
where θ is constant.


## 2. Univalence Conditions forIγk,δ(ai,bj;q;z)

Theorem 7.
Letfk∈𝒜 for all k=1,…,m,γk∈𝒞, and M≥1 with
(24)1Re(δ)∑k=1m‍[(2-μk)M+1]|γk|≤1.
If for all k=1,…,m,fk∈𝔅sr(ai,bj,q,μk),0≤μk<1, and
(25)|ℳ(ai,bj;q)f(z)fk(z)|≤M,(z∈𝒰)
then the integral operator Iγk,δ(ai,bj;q;z) defined by (11) is analytic and univalent in 𝒰.Proof.
From the definition of the operatorℳ(ai,bj;q)f(z)f it can be observed that
(26)ℳ(ai,bj;q)f(z)z≠0,(z∈𝒰),
and for z=0, we have
(27)(ℳ(ai,bj;q)f(z)f1(z)z)1/γ1⋯(ℳ(ai,bj;q)f(z)fm(z)z)1/γm=1.
We define the function h(z) by the form
(28)h(z)=∫0z‍∏k=1m‍(ℳ(ai,bj;q)f(z)fk(t)t)1/γkdt.Therefore(29)h′(z)=∏k=1m‍(ℳ(ai,bj;q)f(z)fk(z)z)1/γk.
Differentiating logarithmically and multiplying by z on both sides of (29)
(30)zh′′(z)h′(z)=∑k=1m‍1γk(z(ℳ(ai,bj;q)f(z)fk(z))′ℳ(ai,bj;q)f(z)fk(z)-1).
Thus we have
(31)|zh′′(z)h′(z)|≤∑k=1m‍1|γk||z(ℳ(ai,bj;q)f(z)fk(z))′ℳ(ai,bj;q)f(z)fk(z)-1|.So(32)1-|z|2Re(δ)Re(δ)|zh′′(z)h′(z)|≤1-|z|2Re(δ)Re(δ)×[∑k=1m‍1|γk|(|z(ℳ(ai,bj;q)f(z)fk(z))′ℳ(ai,bj;q)f(z)fk(z)|+1)]≤1-|z|2Re(δ)Re(δ)×[∑k=1m‍1|γk|(|z2(ℳ(ai,bj;q)f(z)fk(z))′[ℳ(ai,bj;q)f(z)fk(z)]2|×|ℳ(ai,bj;q)f(z)fk(z)z|+1{|z2(ℳ(ai,bj;q)f(z)fk(z))′[ℳ(ai,bj;q)f(z)fk(z)]2|})].
Since |ℳ(ai,bj;q)f(z)fk(z)|≤M,(z∈𝒰,k=1,…,m), and fk∈𝔅sr(ai,bj,q,μk) for all k=1,…,m, then from the Schwarz Lemma and (10), we obtain
(33)1-|z|2Re(δ)Re(δ)|zh′′(z)h′(z)|≤1-|z|2Re(δ)Re(δ)×[(∑k=1m‍1|γk||z2(ℳ(ai,bj;q)f(z)fk(z))′[ℳ(ai,bj;q)f(z)fk(z)]2|M+M+1{|z2(ℳ(ai,bj;q)f(z)fk(z))′[ℳ(ai,bj;q)f(z)fk(z)]2|})]≤1Re(δ)∑k=1m‍1|γk|[(2-μk)M+1],(z∈𝒰)
which, in the light of the hypothesis (24), yields
(34)1-|z|2Re(δ)Re(δ)|zh′′(z)h′(z)|≤1,(z∈𝒰).
Applying Lemma (1) for the function h(z) we obtain that Iγk,δ(ai,bj;q;z) is univalent.Takingμk=0(forallk=1,…,m),M=1,ai=qαi,bj=qβj,q→1, and γk=1/(α-1),δ=1+m(α-1) in Theorem 7, we have the following.Corollary 8 (see [12]).
Letfk∈𝒜 for all k=1,…,m and α∈ℂ with
(35)|α-1|≤Re(α)3m.
If
(36)|z2(Hsr(α1,β1)fk(z))′(Hsr(α1,β1)fk(z))2-1|<1,(z∈𝒰)
and for all k=1,…,m, then the integral operator Fα(α1,β1;z) defined by (12) is analytic and univalent in 𝒰.Takingμk=0(for allk=1,…,m),M=1,r=1,s=0,a1=q, and γk=1/(α-1),δ=1+m(α-1) in Theorem 7, we have the following.Corollary 9.
Letfk∈𝒜 for all k=1,…,m and α∈ℂ with
(37)|α-1|≤Re(α)3m.
If
(38)|z2fk′(z)(fk(z))2-1|<1,(z∈𝒰)
and for all k=1,…,m, then the integral operator Fm,α(z) defined by (13) is analytic and univalent in 𝒰.Theorem 10.
Letfk∈𝒜 for all k=1,…,m,δ,γk∈ℂ, and M≥1 with
(39)|c|≤1-1δ∑k=1m‍[(2-μk)M+1]|γk|,c∈ℂ.
If for all k=1,…,m,fk∈𝔅sr(ai,bj,q,μk),0≤μk<1, and
(40)|ℳ(ai,bj;q)f(z)fk(z)|≤M,(z∈𝒰),
then the integral operator Iγk,δ(ai,bj;q) defined by (11) is analytic and univalent in 𝒰.Proof.
From the proof of Theorem7, we have
(41)zh′′(z)h′(z)=∑k=1m‍1γk(z(ℳ(ai,bj;q)f(z)fk(z))′ℳ(ai,bj;q)f(z)fk(z)-1).
Thus we have
(42)|c|z|2δ+(1-|z|2δ)zh′′(z)δh′(z)|≤|c|+|(1-|z|2δ)zh′′(z)δh′(z)|.
From this result and using the proof of Theorem 7 we obtain
(43)|c|z|2δ+(1-|z|2δ)zh′′(z)δh′(z)|≤|c|+1δ∑k=1m‍1|γk|[(2-μk)M+1].
Since|c|≤1-(1/δ)∑k=1m‍(1/γk)[(2-μk)M+1], then we have
(44)|c|z|2δ+(1-|z|2δ)zh′′(z)δh′(z)|≤1,(z∈𝒰).
Applying Lemma (4) for the function h(z) we obtain that Iγk,δ(ai,bj;q;z) is univalent.Takingμk=0(for allk=1,…,m),r=1,s=0,a1=q, and γk=1/(α-1),δ=1+m(α-1)(α∈R) in Theorem 10, we have the following.Corollary 11.
Letfk∈𝒜 for all k=1,…,m; c∈ℂ,α∈R, and M≥1 with
(45)|c|≤1+(1-α1+m(α-1))(2M+1)m,α∈[1,2Mm+12Mm].
If for all k=1,…,m(46)|z2fk′(z)fk2(z)-1|<1,(z∈𝒰),|fk(z)|≤M,(z∈𝒰;k=1,…,m),
then the integral operator Fm,α(z) defined by (13) is analytic and univalent in 𝒰.Lettingm=1,M=1,andf1=f in Corollary 11, we have the following.Corollary 12.
Letf∈𝒜, c∈ℂandα∈R with
(47)|c|≤3-2αα,(c≠-1),α∈[1,32].
If
(48)|z2f′(z)f2(z)-1|<1,(z∈𝒰),|f(z)|≤1,(z∈𝒰),
then the integral operator Gα(z) defined by (17) is analytic and univalent in 𝒰.Remark 13.
Many other interesting corollaries and results can be obtained by specializing the parameters in Theorem10; for example, see [13, 20, 21].



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