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

We introduce and investigate a new general subclass of analytic and bi-univalent functions in the open unit disk . For functions belonging to this class, we obtain estimates on the first two Taylor-Maclaurin coefficients and . 1. Introduction Let denote the class of all functions of the form which are analytic in the open unit disk We also denote by the class of all functions in the normalized analytic function class which are univalent in . Since univalent functions are one-to-one, they are invertible and the inverse functions need not be defined on the entire unit disk . In fact, the Koebe one-quarter theorem [1] ensures that the image of under every univalent function contains a disk of radius . Thus every function has an inverse , which is defined by In fact, the inverse function is given by Denote by the Hadamard product (or convolution) of the functions and ; that is, if is given by (1) and is given by then For two functions and , analytic in , we say that the function is subordinate to in and write if there exists a Schwarz function , which is analytic in with such that Indeed, it is known that Furthermore, if the function is univalent in , then we have the following equivalence: A function is said to be bi-univalent in if both and are univalent in . Let denote the class of bi-univalent functions in given by (1). For a brief history and interesting examples of functions in the class , 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 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, 8–15]). Throughout this paper, we assume that is an analytic function with positive real part in the unit disk , satisfying , , and 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 and let . We say that if the following conditions are satisfied: where the function is given by Remark 2. If we let then the class reduces to the class denoted by which is the subclass of the functions satisfying where the function is defined by which was introduced and studied recently by Tang et al. [4]. Remark 3. If we let then the class reduces to the new class denoted by which is the subclass of the functions satisfying where the function is