Coefficient Estimates for a New Subclass of Analytic and Bi-Univalent Functions Defined by Hadamard Product

Serap Bulut Civil Aviation College, Kocaeli University, Arslanbey Campus, 41285 İzmit, Kocaeli, Turkey Correspondence should be addressed to Serap Bulut; bulutserap@yahoo.com Received 19 July 2014; Accepted 10 October 2014; Published 10 November 2014 Academic Editor: Jacek Dziok Copyright © 2014 Serap Bulut. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Introduction
Let A denote the class of all functions of the form which are analytic in the open unit disk We also denote by S the class of all functions in the normalized analytic function class A which are univalent in U.
Since univalent functions are one-to-one, they are invertible and the inverse functions need not be defined on the entire unit disk U.In fact, the Koebe one-quarter theorem [1] ensures that the image of U under every univalent function  ∈ S contains a disk of radius 1/4.Thus every function  ∈ A has an inverse  −1 , which is defined by  −1 ( ()) =  ( ∈ U) ,  ( −1 ()) =  (|| <  0 () ;  0 () ≥ 1 4 ) . ( In fact, the inverse function  −1 is given by Denote by  * Θ the Hadamard product (or convolution) of the functions  and Θ; that is, if  is given by (1) and Θ is given by then For two functions  and Θ, analytic in U, we say that the function  is subordinate to Θ in U and write if there exists a Schwarz function , which is analytic in U with such that Indeed, it is known that Furthermore, if the function Θ is univalent in U, then we have the following equivalence: A function  ∈ A is said to be bi-univalent in U if both  and  −1 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 of functions in the class Σ, see [2] (see also [3]).In fact, the aforecited work of Srivastava et al. [2] essentially revived the investigation of various subclasses of the bi-univalent function class Σ in recent years; it was followed by such works as those by Tang et al. [4], El-Ashwah [5], Frasin and Aouf [6], Aouf et al. [7], and others (see, e.g., [2,[8][9][10][11][12][13][14][15]).
Throughout this paper, we assume that  is an analytic function with positive real part in the unit disk U, satisfying  (0) = 1,   (0) > 0, and  (U) is symmetric with respect to the real axis.Such a function has a series expansion of the form With this assumption on , we now introduce the following subclass of bi-univalent functions.Definition 1.Let the function , defined by (1), be in the analytic function class A and let Θ ∈ Σ.We say that if the following conditions are satisfied: where the function ( * Θ) −1 is given by Remark 2. If we let then the class H , Σ (; Θ) reduces to the class denoted by H  Σ (, ) which is the subclass of the functions  ∈ Σ satisfying where the function  is defined by which was introduced and studied recently by Tang et al. [4].
For more results see also [2,8,13,[15][16][17]. Firstly, in order to derive our main results, we need to go to following lemma.Lemma 5 (see [18]).If  ∈ P, then |  | ≤ 2 for each , where P is the family of all functions  analytic in U for which for  ∈ U.

A Set of General Coefficient Estimates
In this section, we state and prove our general results involving the bi-univalent function class H , Σ (; Θ) given by Definition 1. (37)

1 }
; Θ) reduces to the new class denoted by H , Σ (; Θ) which is the subclass of the functions  ∈ Σ satisfying           arg {           <  2 , then the class H , Σ (; Θ) reduces to the new class denoted by H , Σ (; Θ) which is the subclass of the functions  ∈ Σ satisfying