%0 Journal Article
%T On Certain Class of Analytic Functions Related to Cho-Kwon-Srivastava Operator
%A F. Ghanim
%A M. Darus
%J International Journal of Mathematics and Mathematical Sciences
%D 2011
%I Hindawi Publishing Corporation
%R 10.1155/2011/459063
%X Motivated by a multiplier transformation and some subclasses of meromorphic functions which were defined by means of the Hadamard product of the Cho-Kwon-Srivastava operator, we define here a similar transformation by means of the Ghanim and Darus operator. A class related to this transformation will be introduced and the properties will be discussed. 1. Introduction Let denote the class of meromorphic functions normalized by which are analytic in the punctured unit disk . For , we denote by and the subclasses of consisting of all meromorphic functions which are, respectively, starlike of order and convex of order in (cf. e.g., [1¨C4]). For functions defined by we denote the Hadamard product (or convolution) of and by Let us define the function by for , and , where is the Pochhammer symbol. We note that where is the well-known Gaussian hypergeometric function. Let us put Corresponding to the functions and and using the Hadamard product for , we define a new linear operator on by The meromorphic functions with the generalized hypergeometric functions were considered recently by Dziok and Srivastava [5, 6], Liu [7], Liu and Srivastava [8¨C10], and Cho and Kim [11]. For a function , we define and, for , Note that if , , the operator reduced to the one introduced by Cho et al. [12] for . It was known that the definition of the operator was motivated essentially by the Choi-Saigo-Srivastava operator [13] for analytic functions, which includes a simpler integral operator studied earlier by Noor [14] and others (cf. [15¨C17]). Note also the operator has been recently introduced and studied by Ghanim and Darus [18] and Ghanim et al. [19], respectively. To our best knowledge, the recent work regarding operator was charmingly studied by Piejko and Sok¨®l [20]. Moreover, the operator was then defined and studied by Ghanim and Darus [21]. In the same direction, we will study for the operator given in (1.10). Now, it follows from (1.8) and (1.10) that Making use of the operator , we say that a function is in the class if it satisfies the following subordination condition: Furthermore, we say that a function is a subclass of the class of the form The main object of this paper is to present several inclusion relations and other properties of functions in the classes and which we have introduced here. 2. Main Results We begin by recalling the following result (popularly known as Jack's Lemma), which we will apply in proving our first inclusion theorem. Lemma 2.1 (see [Jack's Lemma] [22]). Let the (nonconstant) function be analytic in with . If attains its maximum value on
%U http://www.hindawi.com/journals/ijmms/2011/459063/