Coefficient Estimate of Biunivalent Functions of Complex Order Associated with the Hohlov Operator

We introduce and investigate a new subclass of the function class of biunivalent functions of complex order defined in the open unit disk, which are associated with the Hohlov operator, satisfying subordinate conditions. Furthermore, we find estimates on the Taylor-Maclaurin coefficients and for functions in this new subclass. Several, known or new, consequences of the results are also pointed out. 1. Introduction, Definitions, and Preliminaries Let denote the class of functions of the following form: which are analytic in the open unit disk By we denote the class of all functions in which are univalent in . Some of the important and well-investigated subclasses of the class include, for example, the class of starlike functions of order in and the class of convex functions of order in . It is well known that every function has an inverse , defined by where A function is said to be biunivalent in , if and are univalent in . Let denote the class of biunivalent functions in given by (1). An analytic function is subordinate to an analytic function , written , provided that there is an analytic function defined on with and satisfying . Ma and Minda [1] unified various subclasses of starlike and convex functions for which either of the quantity or is subordinate to a more general superordinate function. For this purpose, they considered an analytic function with positive real part in the unit disk , , , and maps onto a region starlike with respect to 1 and symmetric with respect to the real axis. The class of Ma-Minda starlike functions consists of functions satisfying the subordination . Similarly, the class of Ma-Minda convex functions consists of functions satisfying the subordination . A function is bi-starlike of Ma-Minda type or biconvex of Ma-Minda type, if both and are, respectively, Ma-Minda starlike or convex. These classes are denoted, respectively, by and . In the sequel, it is assumed that is an analytic function with positive real part in the unit disk , satisfying and , and is symmetric with respect to the real axis. Such a function has a series expansion of the form The convolution or Hadamard product of two functions and is denoted by and is defined as where is given by (1) and . Here, in our present investigation, we recall a convolution operator due to Hohlov [2, 3], which indeed is a special case of the Dziok-Srivastava operator [4, 5]. For the complex parameters , , and , the Gaussian hypergeometric function is defined as where is the Pochhammer symbol (or the shifted factorial) defined as follows: For the positive real values , , and ,