New Subclasses of Biunivalent Functions Involving Dziok-Srivastava Operator

We introduce two new subclasses of biunivalent functions which are defined by using the Dziok-Srivastava operator. Furthermore, we find estimates on the coefficients and for functions in these new subclasses. 1. Introduction Let denote the class of all functions of the form which are analytic in the open unit disc . Also let denote the class of all functions in which are univalent in . Some of the important and well-investigated subclasses of the univalent function class include, for example, the class of starlike functions of order in and the class of convex functions of order in . By definition, we have Ding et al. [1] introduced the following class of analytic functions defined as follows: It is easy to see that for . Thus, for , , and hence is univalent class (see [2–4]). It is well known that every function has an inverse , defined by where 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 in the class see [5]. Brannan and Taha [6] (see also [7]) introduced certain subclasses of the bi-univalent function class similar to the familiar subclasses and of starlike and convex functions of order , respectively (see [8]). Thus, following Brannan and Taha [6] (see also [7]), a function is in the class of strongly bi-starlike functions of order if each of the following conditions is satisfied: where is the extension of to . The classes and of bi-starlike functions of order and biconvex functions of order , corresponding, respectively, to the function classes and , were also introduced analogously. For each of the function classes and , they found nonsharp estimates on the first two Taylor-Maclaurin coefficients and (for details, see [6, 7]). For function given by (1) and given by the Hadamard product (or convolution) of and is defined by For complex parameters and , the generalized hypergeometric function is defined by the following infinite series: where is the Pochhammer symbol (or shift factorial) defined, in terms of the Gamma function , by Correspondingly a function is defined by Dziok and Srivastava [9] (see also [10]) considered a linear operator defined by the following Hadamard product: If is given by (1), then we have where To make the notation simple, we write It easily follows from (14) that The linear operator is a generalization of many other linear operators considered earlier. The object of the present paper is to introduce two new subclasses of the bi-univalent functions which are defined by using the