# New Subclasses of Biunivalent Functions Involving Dziok-Srivastava Operator

Authors: M. K. Aouf; R. M. El-Ashwah; Ahmed M. Abd-Eltawab
Journal: ISRN Mathematical Analysis
 (2013)

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## Abstract

We introduce two new subclasses of biunivalent functions which are defined by using the Dziok-Srivastava operator. Furthermore, we find estimates on the coefficientsa2 and a3 for functions in these new subclasses.


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## Body

## 1. Introduction

LetA denote the class of all functions of the form
(1)f(z)=z+∑n=2∞‍anzn,
which are analytic in the open unit disc U={z∈ℂ:|z|<1}. Also let S denote the class of all functions in A which are univalent in U.Some of the important and well-investigated subclasses of the univalent function classS include, for example, the class S*(β) of starlike functions of order β in U and the class K(β) of convex functions of order β in U. By definition, we have
(2)S*(α)={f∈S:Re(zf′(z)f(z))>β,hhhhhhhhhhhhhhh0≤β<1,z∈U(zf′(z)f(z))},K(α)={f∈S:Re(1+zf′′(z)f′(z))>β,hhhhhhhhhhhhhhhhhh0≤β<1,z∈U(zf′(z)f(z))}.Ding et al. [1] introduced the following class Qλ(β) of analytic functions defined as follows:
(3)Qλ(β)={f∈A:Re((1-λ)f(z)z+λf′(z))>β,hhhhhhhhhhhhhhhhhhhhhhhhhh0≤β<1,λ≥0((1-λ)f(z)z+λf′(z))}.It is easy to see thatQλ1(β)⊂Qλ2(β) for λ1>λ2≥0. Thus, for λ≥1, 0≤β<1, Qλ(β)⊂Q1(β)={f∈A:Ref′(z)>β,0≤β<1} and hence Qλ(β) is univalent class (see [2–4]).It is well known that every functionf∈S has an inverse f-1, defined by
(4)f-1(f(z))=z(z∈U),f(f-1(w))=w(|w|<r0(f);r0(f)≥14),
where
(5)f-1(w)=w-a2w2+(2a22-a3)w3f-1(w)=-(5a23-5a2a3+a4)w4+⋯.A functionf∈A is said to be bi-univalent in U if both f(z) and f-1(z) are univalent in U. Let Σ denote the class of bi-univalent functions in U given by (1). For a brief history and interesting examples in the class Σ see [5].Brannan and Taha [6] (see also [7]) introduced certain subclasses of the bi-univalent function class Σ similar to the familiar subclasses S*(β) and K(β) of starlike and convex functions of order β(0≤β<1), respectively (see [8]). Thus, following Brannan and Taha [6] (see also [7]), a function f∈A is in the class SΣ*(α) of strongly bi-starlike functions of order α(0<α≤1) if each of the following conditions is satisfied:
(6)f∈Σ,|arg(zf′(z)f(z))|<απ2(0<α≤1,z∈U),f∈Σ,|arg(zg′(w)g(w))|<απ2(0<α≤1,z∈U),
where g is the extension of f-1 to U. The classes SΣ*(α) and KΣ(α) of bi-starlike functions of order α and biconvex functions of order α, corresponding, respectively, to the function classes S*(β) and K(β), were also introduced analogously. For each of the function classes SΣ*(α) and KΣ(α), they found nonsharp estimates on the first two Taylor-Maclaurin coefficients |a2| and |a3| (for details, see [6, 7]).For functionf given by (1) and g given by
(7)g(z)=z+∑n=2∞‍bnzn,
the Hadamard product (or convolution) of f and g is defined by
(8)(f*g)(z)=z+∑n=2∞‍anbnzn=(g*f)(z).For complex parametersa1,…,aqand b1,…,bs(bj∉ℤ0-={0,-1,-2,…};j=1,…,s), the generalized hypergeometric function qFs is defined by the following infinite series:
(9)qFs(a1,…,aq;b1,…,bs;z)=∑n=0∞(a1)n⋯(aq)n(b1)n⋯(bs)nznn!(q≤s+1;q,s∈ℕ0=ℕ∪{0},ℕ={1,2,3,…};z∈U),
where (θ)n is the Pochhammer symbol (or shift factorial) defined, in terms of the Gamma function Γ, by
(10)(θ)n=Γ(θ+n)Γ(θ)={1,(n=0)θ(θ+1)⋯(θ+n-1),(n∈ℕ).
Correspondingly a function h(a1,…,aq;b1,…,bs;z) is defined by
(11)h(a1,…,aq;b1,…,bs;z)=zqFs(a1,…,aq;b1,…,bs;z)(z∈U).
Dziok and Srivastava [9] (see also [10]) considered a linear operator
(12)H(a1,…,aq;b1,…,bs):A⟶A,
defined by the following Hadamard product:
(13)H(a1,…,aq;b1,…,bs)f(z)=h(a1,…,aq;b1,…,bs;z)*f(z),(q≤s+1;q,s∈ℕ0;z∈U).
If f∈Ais given by (1), then we have
(14)H(a1,…,aq;b1,…,bs)f(z)=z+∑n=2∞‍Γn[a1;b1]anzn(z∈U),
where
(15)Γn[a1;b1]=(a1)n⋯(aq)n(b1)n⋯(bs)n1n!(n∈ℕ).
To make the notation simple, we write
(16)Hq,s[a1;b1;z]=H(a1,…,aq;b1,…,bs)f(z).
It easily follows from (14) that
(17)z(Hq,s[a1;b1;z])′=a1Hq,s[a1+1;b1;z]-(a1-1)Hq,s[a1;b1;z].
The linear operator Hq,s[a1;b1;z] is a generalization of many other linear operators considered earlier.The object of the present paper is to introduce two new subclasses of the bi-univalent functions which are defined by using the Dziok-Srivastava operator and find estimates on the coefficients|a2| and |a3| for functions in these new subclasses of the function class Σ employing the techniques used earlier by Srivastava et al. [5] (see also [11]).In order to derive our main results, we have to recall here the following lemma [12].Lemma 1.
Ifh∈P, then |ck|≤2 for each k, where P is the family of all functions h analytic in U for which Reh(z)>0h(z)=1+c1z+c2z2+c3z3+⋯ for z∈U.Unless otherwise mentioned, we assume throughout this paper thatai,bj∈ℂ∖ℤ0-,i=1,…,s, j=1,…,q, q≤s+1; q,s∈ℕ0, 0<α≤1, λ≥1, z∈U, Γn[a1;b1] is given by (15) and all powers are understood as principle values.


## 2. Coefficient Bounds of the Function ClassTq,sΣ[a1;b1,α,λ]

Definition 2.
One says that a functionf(z) given by (1) is said to be in the class Tq,sΣ[a1;b1,α,λ] if it satisfies the following condition:
(18)f∈Σ,|arg((1-λ)Hq,s[a1;b1;z]z+λ(Hq,s[a1;b1;z])′Hq,s[a1;b1;z]z)|<απ2,|arg((1-λ)g(w)w+λg′(w))|<απ2,
where the functiong is given by
(19)g(w)=Hq,s-1[a1;b1;z]g(w)=w-Γ2[a1;b1]a2w2g(w)=+(2(Γ2[a1;b1])2a22-Γ3[a1;b1]a3)w3g(w)=-(5(Γ2[a1;b1])3a23-5Γ2[a1;b1]g(w)=×Γ3[a1;b1]a2a3+Γ4[a1;b1]a4)w4+⋯.Remark 3.
(i) Forq=2, s=1, and a1=a2=b1=1, we have T2,1Σ[1,1;2;α,λ]=BΣ(α,λ), where the class BΣ(α,λ) was introduced and studied by Frasin and Aouf [11].
(ii) Forq=2, s=1, and a1=a2=b1=λ=1, we have T2,1Σ[1,1;2;α,1]=HΣ(α,λ), where the class HΣ(α,λ) was introduced and studied by Srivastava et al. [5].Theorem 4.
Lettingf(z) given by (1) be in the class Tq,sΣ[a1;b1,α,λ], then
(20)|a2|=2α|Γ2[a1;b1]|(λ+1)2+α(1+2λ-λ2),(21)|a3|=4α2|Γ3[a1;b1]|(λ+1)2+2α|Γ3[a1;b1]|(2λ+1).Proof.
It follows from (18) that
(22)(1-λ)Hq,s[a1;b1;z]z+λ(Hq,s[a1;b1;z])′=[p(z)]2,(1-λ)g(w)w+λg′(w)=[q(w)]2,
where p(z) and q(w) in P have the forms
(23)p(z)=1+p1z+p2z2+p3z3+⋯,(24)q(w)=1+q1w+q2w2+q3w3+⋯.
Now, equating the coefficients in (22), we get
(25)(λ+1)Γ2[a1;b1]a2=αp1,(26)(2λ+1)Γ3[a1;b1]a3=αp2+α(α-1)2p12,(27)-(λ+1)Γ2[a1;b1]a2=αq1,(28)(2λ+1)(2(Γ2[a1;b1])2a22-Γ3[a1;b1]a3)=αq2+α(α-1)2q12.
From (25) and (27), we get
(29)p1=-q1,(30)2(λ+1)2(Γ2[a1;b1])2a22=α2(p12+q12).
Now from (26), (28), and (30), we obtain
(31)2(2λ+1)(Γ2[a1;b1])2a22=α(p2+q2)+α(α-1)2(p12+q12)=α(p2+q2)+α(α-1)22(λ+1)2(Γ2[a1;b1])2a22α2.
Therefore, we have
(32)a22=α2(p2+q2)(Γ2[a1;b1])2[(λ+1)2+α(1+2λ-λ2)].
Applying Lemma 1 for the coefficients p2 and q2, we immediately have
(33)|a2|≤2α|Γ2[a1;b1]|(λ+1)2+α(1+2λ-λ2).
This gives the bound on |a2| as asserted in (20).
Next, in order to find the bound on|a3|, by subtracting (28) from (26) and using (29), we get
(34)2(2λ+1)Γ3[a1;b1]a3-2(2λ+1)(Γ2[a1;b1])2a22=αp2+α(α-1)2p12-(αq2+α(α-1)2q12)=α(p2-q2).
It follows from (30) and (34) that
(35)2(2λ+1)Γ3[a1;b1]a3=α2(2λ+1)(p12+q12)(λ+1)2+α(p2-q2),
And, then,
(36)a3=α2(p12+q12)2(λ+1)2Γ3[a1;b1]+α(p2-q2)2(2λ+1)Γ3[a1;b1].
Applying Lemma 1 once again for the coefficients p1, p2, q1, and q2, we readily get
(37)|a3|≤4α2(λ+1)2|Γ3[a1;b1]|+2α(2λ+1)|Γ3[a1;b1]|.
This completes the proof of Theorem4.Remark 5.
(i) Takingq=2, s=1, and a1=a2=b1=1, in Theorem 4, we obtain the result obtained by Frasin and Aouf [11, Theorem 2.2].
(ii) Takingq=2, s=1, and a1=a2=b1=λ=1, in Theorem 4, we obtain the result obtained by Srivastava et al. [5, Theorem 1].


## 3. Coefficient Bounds of the Function ClassTq,sΣ[a1;b1,β,λ]

Definition 6.
One says that a functionf(z) given by (1) is said to be in the class Tq,sΣ[a1;b1,β,λ] if it satisfies the following condition:
(38)f∈Σ,Re{(1-λ)Hq,s[a1;b1;z]z+λ(Hq,s[a1;b1;z])′Hq,s[a1;b1;z]z}>β,Re{(1-λ)g(w)w+λg′(w)}>β,
where the function g is defined by (19).Remark 7.
(i) Forq=2, s=1, and a1=a2=b1=1, we have T2,1Σ[1,1;2;β,λ]=BΣ(β,λ), where the class BΣ(β,λ) was introduced and studied by Frasin and Aouf [11].
(ii) Forq=2, s=1, and a1=a2=b1=λ=1, we have T2,1Σ[1,1;2;β,1]=HΣ(β,λ), where the class HΣ(β,λ) was introduced and studied by Srivastava et al. [5].Theorem 8.
Lettingf(z) given by (1) be in the class Tq,sΣ[a1;b1,β,λ], 0≤β<1 and λ≥1, then
(39)|a2|=2(1-β)|Γ2[a1;b1]|2λ+1,(40)|a3|=4(1-β)2|Γ3[a1;b1]|(λ+1)2+2(1-β)|Γ3[a1;b1]|(2λ+1).Proof.
It follows from (38) that
(41)(1-λ)Hq,s[a1;b1;z]z+λ(Hq,s[a1;b1;z])′=β+(1-β)p(z),(1-λ)g(w)w+λg′(w)=β+(1-β)q(w),
where p(z) and q(w) have the forms (23) and (24), respectively.
As in the proof of Theorem4, by suitably comparing coefficients in (41), we get
(42)(λ+1)Γ2[a1;b1]a2=(1-β)p1,(43)(2λ+1)Γ3[a1;b1]a3=(1-β)p2,(44)-(λ+1)Γ2[a1;b1]a2=(1-β)q1,(45)(2λ+1)(2(Γ2[a1;b1])2a22-Γ3[a1;b1]a3)=(1-β)q2.
From (42) and (44), we get
(46)p1=-q1,(47)2(λ+1)2(Γ2[a1;b1])2a22=(1-β)2(p12+q12).
Also, from (43) and (45), we find that
(48)2(2λ+1)(Γ2[a1;b1])2a22=(1-β)(p2+q2).
Therefore, we have
(49)|a22|≤(1-β)(Γ2[a1;b1])2[2(2λ+1)](|p2|+|q2|).
Applying Lemma 1 for the coefficients p2 and q2, we immediately have
(50)|a2|≤2(1-β)|Γ2[a1;b1]|2λ+1.
This gives the bound on |a2| as asserted in (39).
Next, in order to find the bound on|a3|, by subtracting (45) from (43), we get
(51)2(2λ+1)Γ3[a1;b1]a3-2(2λ+1)(Γ2[a1;b1])2a22=(1-β)(p2-q2),
or, equivalently,
(52)a3=(Γ2[a1;b1])2a22Γ3[a1;b1]+(1-β)(p2-q2)2(2λ+1)Γ3[a1;b1],
and, then from (47), we find that
(53)a3=(1-β)2(p12+q12)2(λ+1)2Γ3[a1;b1]+(1-β)(p2-q2)2(2λ+1)Γ3[a1;b1].
Applying Lemma 1 once again for the coefficients p1, p2, q1, and q2, we readily get
(54)|a3|≤4(1-β)2(λ+1)2|Γ3[a1;b1]|+2(1-β)(2λ+1)|Γ3[a1;b1]|.
This completes the proof of Theorem 8.Remark 9.
(i) Takingq=2, s=1, and a1=a2=b1=1, in Theorem 8, we obtain the result obtained by Frasin and Aouf [11, Theorem 3.2].
(ii) Takingq=2, s=1, and a1=a2=b1=λ=1, in Theorem 8, we obtain the result obtained by Srivastava et al. [5, Theorem 2].



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