Coefficient Estimates for a Subclass of Bi-univalent Functions

In this paper, we consider a subclass SΣ(α,β ) of bi-univalent functions defined in the open unit disk D = {z ∈ C : |z|< 1}. Besides, we find upper bounds for the second and third coefficients for functions in this subclass.


Introduction
Let A be a class of functions of the fom Since univalent functions are one-to-one, they are invertible and inverse functions need not be defined on the entire unit disk D. However, the famous Koebe one-quarter theorem [3] ensures that the image of the unit disk D under every function f ∈ S contains a disk of radius 1 4 .Thus, every function f ∈ S has an inverse f −1 defined by and ), where Recently, Srivastava et al. [5] and Frasian and Aouf [4] and Caglar et al. [2] have introduced and have investigated various subclasses of bi-univalent functions and found estimates on the coefficients |a 2 | and |a 3 | for functions in these classes.In this paper we introduce a subclass S Σ (α, β ) of bi-univalent functions and obtain estimates on the coefficients |a 2 | and |a 3 | for functions in this subclass.

Preliminaries and notations
In order to derive our main results, we need to following lemma [3]: Re and Re where g(w) = f −1 (w) is given by (1.2).
Next, in order to find the bound on |a 3 |, by subtracting (3.12) from (3.10), we get which, upon substitution of the value a 2 2 from (3.14), yields On the other hand, by using the equation (3.15) in (3.18), we obtain

. 1 )
which are analytic in the open unit disk D = {z ∈ C : |z| < 1}.Also, let S denote a subclass of all functions in A which are univalent in D.

Lemma 2 . 1 .
If p ∈ P, then |p k | 2 for each k, where P is the family of all functions p analytic in D for which Rep(z) > 0, p(z) = 1 + p 1 z + p 2 z 2 + . . .for z ∈ D. Definition 2.1.A function f (z) ∈ Σ given by (1.1) is said to be in class S Σ (α, β ) if the following conditions are satisfied: