# Coefficient Bounds for Certain Subclasses of Bi-Univalent Function

Authors: G. Murugusundaramoorthy; N. Magesh; V. Prameela
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We introduce two new subclasses of the function class Σ of bi-univalent functions defined in the open unit disc. Furthermore, we find estimates on the coefficients|a2| and |a3| for functions in these new subclasses. Also consequences of the results are pointed out.


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## Body

## 1. Introduction

Denote by𝒜 the class of functions of the form
(1)f(z)=z+∑n=2∞anzn
which are analytic in the open unit disc 𝕌={z:z∈ℂand|z|<1}. Further, denote by 𝒮 the class of all functions in 𝒜 which are univalent and normalized by f(0)=0=f′(0) in 𝕌. The well-investigated subclasses of the univalent function class 𝒮 are the class of starlike functions of order α(0≤α<1), denoted by 𝒮*(α) and the class of convex functions of order α denoted by 𝒦(α) in 𝕌. It is well known that every function f∈𝒮 has an inverse f-1, defined by
(2)f-1(f(z))=z,z∈𝕌,f(f-1(w))=w,|w|<r0(f),r0(f)≥14,
where
(3)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).Analogous to the function class𝒮, the bi-univalent function class Σ include, for example, the class 𝒮Σ*(α) of bi-starlike functions of order α  (0≤α<1), the class 𝒦Σ(α) of biconvex functions of order α  (0≤α<1), and the class 𝒮Σα of strongly bi-starlike functions of order α  (0<α≤1). For some intriguing examples of functions and characterization of the class Σ, one could refer to Srivastava et al. [1] and the references stated therein (see also [2]). Recently there has been triggering, interest to study the bi-univalent functions class Σ (see [2–5]) and obtain nonsharp estimates on the first two Taylor-Maclaurin coefficients |a2| and |a3|. The coefficient estimate problem for each of the following Taylor-Maclaurin coefficients |an| for n∈ℕ∖{1,2}, ℕ:={1,2,3,…} is presumably still an open problem.Motivated by the earlier works of Srivastava et al. [1] and Frasin and Aouf [3] in the present paper we introduce the following two new subclasses of the function class Σ.Definition 1.
A functionf(z) given by (1) is said to be in the class 𝒢Σ(α,λ) if the following conditions are satisfied:
(4)f∈Σ,|arg(zf′(z)(1-λ)f(z)+λzf′(z))|<απ2,hhhhcfgghhcch0<α≤1,0≤λ<1,z∈𝕌,|arg(wg′(w)(1-λ)g(w)+λwg′(w))|<απ2,hvvvhi0<α≤1,0≤λ<1,w∈𝕌,
where the function g is given by
(5)g(w)=w-a2w2+(2a22-a3)w3g(w)-(5a23-5a2a3+a4)w4+⋯.Definition 2.
A functionf(z) given by (1) is said to be in the class ℳΣ(β,λ) if the following conditions are satisfied:
(6)f∈Σ,ℜ(zf′(z)(1-λ)f(z)+λzf′(z))>β,hhhhhhhhi0≤β<1,0≤λ<1,z∈𝕌,ℜ(wg′(w)(1-λ)g(w)+λwg′(w))>β,jjjj0≤β<1,0≤λ<1,w∈𝕌,
where the function g is given by (5).
It is of interest to note that, forλ=0, the class 𝒢Σ(α,λ) reduces to 𝒮Σα of strongly bi-starlike functions of order α and the class ℳΣ(β,λ) leads to 𝒮Σ*(β) bi-starlike functions of order β.
The object of the present paper is to find estimates on the coefficients|a2| and |a3| for functions in the above-defined subclasses 𝒢Σ(α,λ) and ℳΣ(β,λ) of the function class Σ by employing the techniques used earlier by Srivastava et al. [1].
In order to derive our main results, we recall the following lemma.Lemma 3 (see [6]).
Ifh∈𝒫, then |ck|≤2 for each k, where 𝒫, is the family of all functions h analytic in 𝕌 for which ℜ{h(z)}>0, where h(z)=1+c1z+c2z2+⋯ for z∈𝕌.


## 2. Coefficient Bounds for the Function Class𝒢Σ(α,λ)

We begin by finding the estimates on the coefficients|a2| and |a3| for functions in the class f∈𝒢Σ(α,λ).Theorem 4.
Letf(z) given by (1) be in the class 𝒢Σ(α,λ),  0<α≤1, and 0≤λ<1. Then
(7)|a2|≤2α(1-λ)1+α,(8)|a3|≤4α2(1-λ)2+α1-λ.Proof.
It follows from (4) that
(9)zf′(z)(1-λ)f(z)+λzf′(z)=[p(z)]α,wg′(w)(1-λ)g(w)+λwg′(w)=[q(w)]α,
where p(z) and q(w) in 𝒫 have the forms
(10)p(z)=1+p1z+p2z2+⋯,(11)q(z)=1+q1w+q2w2+⋯.
Now, equating the coefficients in (9), we get
(12)(1-λ)a2=αp1,(13)(λ2-1)a22+2(1-λ)a3=12[α(α-1)p12+2αp2],(14)-(1-λ)a2=αq1,(15)(λ2-4λ+3)a22-2(1-λ)a3=12[α(α-1)q12+2αq2].
From (12) and (14), we get
(16)p1=-q1,(17)2(1-λ)2a22=α2(p12+q12).
From (13), (15), and (17), we obtain
(18)a22=α2(p2+q2)(α+1)(1-λ)2.
Applying Lemma 3 for the coefficients p2 and q2, we immediately have
(19)|a2|≤2α(1-λ)(1+α).
This gives the bound on |a2| as asserted in (7).
Next, in order to find the bound on|a3|, by subtracting (15) from (13), we get
(20)4(1-λ)a3-4(1-λ)a22=α(p2-q2)+α(α-1)2(p12-q12).
It follows from (16), (17), and (20) that
(21)a3=α(p2-q2)4(1-λ)+α2(p12+q12)2(1-λ)2.
Applying Lemma 3 once again for the coefficients p1,p2,q1,and q2, we readily get
(22)|a3|≤4α2(1-λ)2+α1-λ.
This completes the proof of Theorem 4.In the following section we find the estimates on the coefficients|a2| and |a3| for functions in the class ℳΣ(β,λ).


## 3. Coefficient Bounds for the Function ClassℳΣ(β,λ)

Theorem 5.
Letf(z) given by (1) be in the class ℳΣ(β,λ),0≤β<1, and 0≤λ<1. Then
(23)|a2|≤2(1-β)1-λ,|a3|≤4(1-β)2(1-λ)2+(1-β)1-λ.Proof.
It follows from (6) that there exists p,q∈𝒫 such that
(24)zf′(z)(1-λ)f(z)+λzf′(z)=β+(1-β)p(z),wg′(w)(1-λ)g(w)+λwg′(w)=β+(1-β)q(w),
where p(z) and q(w) have the forms of (10) and (11), respectively. Equating coefficients in (24) we get
(25)(1-λ)a2=(1-β)p1,(λ2-1)a22+2(1-λ)a3=(1-β)p2,-(1-λ)a2=(1-β)q1,(λ2-4λ+3)a22-2(1-λ)a3=(1-β)q2.
The proof follows, by employing the techniques used in the proof of Theorem4.Takingλ=0 in Theorems 4 and 5 one can get the following corollaries.Corollary 6.
Letf(z) given by (1) be in the class 𝒮Σα and 0<α≤1. Then
(26)|a2|≤2αα+1,|a3|≤4α2+α.Corollary 7.
Letf(z) given by (1) be in the class 𝒮Σ*(β) and 0≤β<1. Then
(27)|a2|≤2-2β,|a3|≤4(1-β)2+(1-β).



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