A Subclass of Harmonic Univalent Functions Associated with -Analogue of Dziok-Srivastava Operator

We study a class of complex-valued harmonic univalent functions using a generalized operator involving basic hypergeometric function. Precisely, we give a necessary and sufficient coefficient condition for functions in this class. Distortion bounds, extreme points, and neighborhood of such functions are also considered. 1. Introduction Let be the open unit disc, and let denote the class of functions which are complex valued, harmonic, univalent, and sense preserving in normalized by . Each can be expressed as , where and are analytic in . We call the analytic part and the coanalytic part of . A necessary and sufficient condition for to be locally univalent and sense preserving in is that in (see [1]). In [2], there is a more comprehensive study on harmonic univalent functions. Thus, for , we may write Note that reduces to , the class of normalized analytic univalent functions, if the coanalytic part of is identically zero. The study of basic hypergeometric series (also called -hypergeometric series) essentially started in 1748 when Euler considered the infinite product . In the literature, we were told that the development of these functions was much slower until, in 1878, Heine converted a simple observation that which returns the theory of basic hypergeometric series to the famous theory of Gauss’s hypergeometric series. The importance of basic hypergeometric functions is due to their application in deriving -analogue of well-known functions, such as -analogues of the exponential, gamma, and beta functions. In this paper, we define a class of starlike harmonic functions using basic hypergeometric functions and investigate its properties like coefficient condition, distortion theorem, and extreme points. For complex parameters , , , , , , we define the basic hypergeometric function by where denote the set of positive integers and is the -shifted factorial defined by We note that where is the well-known generalized hypergeometric function. By the ratio test, one observes that for and the series defined in (2) converges absolutely in so that it represented an analytic function in . For more mathematical background of basic hypergeometric functions, one may refer to [3, 4]. The -derivative of a function is defined by For a function , we can observe that Then , where is the ordinary derivative. For more properties of , see [4, 5]. Corresponding to the function , consider The authors [6] defined the linear operator by where stands for convolution and To make the notation simple, we write We define the operator (8) of harmonic function given by (1) as