# Coefficient Estimates for Certain Classes of Bi-Univalent Functions

Authors: Jay M. Jahangiri; Samaneh G. Hamidi
Journal: International Journal of Mathematics and Mathematical Sciences
 (2013)

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## Abstract

A function analytic in the open unit disk𝔻 is said to be bi-univalent in 𝔻 if both the function
and its inverse map are univalent there. The bi-univalency condition imposed
on the functions analytic in 𝔻 makes the behavior of their
coefficients unpredictable. Not much is known about the behavior of the
higher order coefficients of classes of bi-univalent functions. We use Faber
polynomial expansions of bi-univalent functions to obtain estimates for
their general coefficients subject to certain gap series as well as
providing bounds for early coefficients of such functions.


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## Body

## 1. Introduction

Let𝒜 denote the class of functions f which are analytic in the open unit disk 𝔻:={z∈ℂ:|z|<1} and normalized by
(1)f(z)=z+∑n=2∞‍anzn.Let𝒮 denote the class of functions f∈𝒜 that are univalent in 𝔻 and let 𝒫 be the class of functions p(z)=1+∑n=1∞‍pnzn that are analytic in 𝔻 and satisfy the condition Re(p(z))>0 in 𝔻. By the Caratheodory lemma (e.g., see [1]) we have |pn|≤2.For0≤α<1 and λ≥1 we let 𝒟(α;λ) denote the family of analytic functions f∈𝒜 so that
(2)Re((1-λ)f(z)z+λf′(z))>α,z∈𝔻.We note that𝒟(0;1) is the class of bounded boundary turning functions and also that 𝒟(α;λ)⊂𝒟(β;λ) if 0≤β<α. For f∈𝒜, the class 𝒟(α;λ)⊂𝒮 and was first defined and investigated by Ding et al. [2].It is well known that every functionf∈𝒮  has an inverse f-1 satisfying f-1(f(z))=z for z∈𝔻 and f(f-1(w))=w for |w|<1/4, according to Kobe One Quarter Theorem (e.g., see [1]).A functionf∈𝒜 is said to be bi-univalent in 𝔻 if both f∈𝒮 and g=f-1∈𝒮. Finding bounds for the coefficients of classes of bi-univalent functions dates back to 1967 (see Lewin [3]). But the interest on the bounds for the coefficients of classes of bi-univalent functions picked up by the publications of Brannan and Taha [4], Srivastava et al. [5], Frasin and Aouf [6], Ali et al. [7], and Hamidi et al. [8]. The bi-univalency condition imposed on the functions f∈𝒜 makes the behavior of their coefficients unpredictable. Not much is known about the behavior of the higher order coefficients of classes of bi-univalent functions, as Ali et al. [7] also remarked that finding the bounds for |an| when n≥4 is an open problem. Here in this paper we let f∈𝒟(α;λ) and g=f-1∈𝒟(α;λ) and use the Faber polynomial coefficient expansions to provide bounds for the general coefficients |an| of such functions with a given gap series. We also obtain estimates for the first two coefficients |a2| and |a3| of these functions as well as providing an estimate for their coefficient body (a2,a3). The bounds provided in this paper prove to be better than those estimates provided by Srivastava et al. [5] and Frasin and Aouf [6].


## 2. Main Results

Using the Faber polynomial expansion of functionsf∈𝒜 of the form (1), the coefficients of its inverse map g=f-1 may be expressed as, [9],
(3)g(w)=f-1(w)=w+∑n=2∞‍1nKn-1-n(a2,a3,…)wn,
where
(4)Kn-1-n=(-n)!(-2n+1)!(n-1)!a2n-1+(-n)!(2(-n+1))!(n-3)!a2n-3a3+(-n)!(-2n+3)!(n-4)!a2n-4a4+(-n)!(2(-n+2))!(n-5)!a2n-5[a5+(-n+2)a32]+(-n)!(-2n+5)!(n-6)!a2n-6[a6+(-2n+5)a3a4]+∑j≥7‍a2n-jVj,
such that Vj with 7≤j≤n is a homogeneous polynomial in the variables a2,a3,…,an [10]. In particular, the first three terms of Kn-1-n are
(5)12K1-2=-a2,13K2-3=2a22-a3,14K3-4=-(5a23-5a2a3+a4).In general, for anyp∈ℕ, an expansion of Knp is as, [9, page 183],
(6)Knp=pan+p(p-1)2Dn2+p!(p-3)!3!Dn3+⋯+p!(p-n)!n!Dnn,
where Dnp=Dnp(a2,a3,…) and by [11] or [12],
(7)Dnm(a1,a2,…,an)=∑m=1∞‍m!(a1)μ1⋯(an)μnμ1!⋯μn!,
while a1=1, and the sum is taken over all nonnegative integers μ1,…,μn satisfying
(8)μ1+μ2+⋯+μn=m,μ1+2μ2+⋯+nμn=n.Evidently,Dnn(a1,a2,…,as+m)=a1n, [13].Theorem 1.
For0≤α<1 and λ≥1 let f∈𝒟(α;λ) and g∈𝒟(α;λ). If ak=0; 2≤k≤n-1, then
(9)|an|≤2(1-α)1+(n-1)λ;n≥4.Proof.
For analytic functionsf of the form (1) we have
(10)(1-λ)f(z)z+λf′(z)=1+∑n=2∞‍(1+(n-1)λ)anzn-1,
and for its inverse map, g=f-1, we have
(11)(1-λ)g(w)w+λg′(w)=1+∑n=2∞‍(1+(n-1)λ)bnwn-1=1+∑n=2∞‍(1+(n-1)λ)fffffffffffffffffffff×1nKn-1-n(a2,a3,…,an)wn-1.
On the other hand, sincef∈𝒟(α;λ) and g=f-1∈𝒟(α;λ), by definition, there exist two positive real part functions p(z)=1+∑n=1∞‍cnz-n and q(w)=1+∑n=1∞‍dnw-n where Rep(z)>0 and Req(w)>0 in 𝒟 so that
(12)(1-λ)f(z)z+λf′(z)=1+(1-α)∑n=1∞‍Kn1(c1,c2,…,cn)zn,(13)(1-λ)g(w)w+λg′(w)=1+(1-α)∑n=1∞‍Kn1(d1,d2,…,dn)wn.
Comparing the corresponding coefficients of (10) and (12) yields
(14)(1+λ(n-1))an=(1-α)Kn-11(c1,c2,…,cn-1),
and similarly, from (11) and (13) we obtain
(15)1n(1+(n-1)λ)Kn-1-n(b0,b1,…,bn)=(1-α)Kn-11(d1,d2,…,dn-1).
Note that forak=0; 2≤k≤n-1 we have bn=-an and so
(16)(1+(n-1)λ)an=(1-α)cn-1,-(1+(n-1)λ)an=(1-α)dn-1.
Now taking the absolute values of either of the above two equations and applying the Caratheodory lemma, we obtain(17)|an|≤(1-α)|cn-1||1+(n-1)λ|=(1-α)|dn-1||1+(n-1)λ|≤2(1-α)1+(n-1)λ.Theorem 2.
For0≤α<1 and λ≥1 let f∈𝒟(α;λ) and g∈𝒟(α;λ). Then one has the following
(18)(i)|a2|≤{2(1-α)1+2λ,0≤α<1+2λ-λ22(1+2λ);2(1-α)1+λ,1+2λ-λ22(1+2λ)≤α<1.(ii)|a3|≤2(1-α)1+2λ.(iii)|a3-2a22|≤2(1-α)1+2λ.Proof.
Replacingn by 2 and 3 in (14) and (15), respectively, we deduce
(19)(1+λ)a2=(1-α)c1,(20)(1+2λ)a3=(1-α)c2,(21)-(1+λ)a2=(1-α)d1,(22)(1+2λ)(2a22-a3)=(1-α)d2.
Dividing (19) or (21) by (1+λ), taking their absolute values, and applying the Caratheodory lemma, we obtain
(23)|a2|≤(1-α)|c1|1+λ=(1-α)|d1|1+λ≤2(1-α)1+λ.
Adding (20) to (22) implies
(24)2(1+2λ)a22=(1-α)(c2+d2)
or
(25)a22=(1-α)(c2+d2)2(1+2λ).
An application of Caratheodory lemma followed by taking the square roots yields(26)|a2|≤(1-α)(|c2|+|d2|)2(1+2λ)2≤2(1-α)1+2λ2.
Now the bounds given in Theorem2 (i) for |a2| follow upon noting that if (1+2λ-λ2)/2(1+2λ)≤α<1, then
(27)2(1-α)1+λ≤2(1-α)1+2λ2.
Dividing (20) by (1+2λ), taking the absolute values of both sides, and applying the Caratheodory lemma yield
(28)|a3|=(1-α)|c2|1+2λ≤2(1-α)1+2λ.
Dividing (22) by (1+2λ), taking the absolute values of both sides, and applying the Caratheodory lemma, we obtain
(29)|a3-2a22|≤2(1-α)1+2λ.Corollary 3.
For0≤α<1 let f∈𝒟(α;1) and g∈𝒟(α;1). Then one has the following
(30)(i)|a2|≤{2(1-α)3,0≤α<13;1-α,13≤α<1.(ii)|a3|≤2(1-α)3.Remark 4.
The above two estimates for|a2| and |a3| show that the bounds given in Theorem 2 are better than those given by Srivastava et al. ([5, page 1191, Theorem 2] and Frasin and Aouf [6, page 1572, Theorem 3.2]).



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