Univalence of a New General Integral Operator Associated with the -Hypergeometric Function

Motivated by the familiar -hypergeometric functions, we introduce a new family of integral operators and obtain new sufficient conditions of univalence criteria. Several corollaries and consequences of the main results are also pointed out. 1. Introduction Let denote the class of functions of the form which are analytic in the open unit disk , and the class of functions which are univalent in . Let , where is defined by (1) and is given by Then the Hadamard product (or convolution) of the functions and is defined by For complex parameters and , we define the -hypergeometric function by , where denotes the set of positive integers and is the -shifted factorial defined by By using the ratio test, we should note that, if , the series (4) converges absolutely for if . For more mathematical background of these functions, one may refer to [1]. Corresponding to the function defined by (4), consider Recently, the authors [2] defined the linear operator by where It should be remarked that the linear operator (7) is a generalization of many operators considered earlier. For , and , we obtain the Dziok-Srivastava linear operator [3] (for ), so that it includes (as its special cases) various other linear operators introduced and studied by Ruscheweyh [4], Carlson and Shaffer [5] and the Bernardi-Libera-Livingston operators [6–8]. The -difference operator is defined by where is the ordinary derivative. For more properties of see [9, 10]. Lemma 1 (see [2]). Let ; then (i) for , and , one has . (ii) For , and , one has and , where is the -derivative defined by (9). Definition 2. A function is said to be in the class if it is satisfying the condition where is the operator defined by (7). Note that , where the class of analytic and univalent functions was introduced and studied by Frasin and Darus [11]. Using the operator , we now introduce the following new general integral operator. For , , and , we define the integral operator by where . Remark 3. It is interesting to note that the integral operator generalizes many operators introduced and studied by several authors, for example, (1) for , and , where and , we obtain the following integral operator introduced and studied by Selvaraj and Karthikeyan [12]: where for convenience , and is the Dziok-Srivastava operator [3]. (2) For , and , we obtain the integral operator studied recently by Breaz et al. [13]. (3) For , and , we obtain the integral operator introduced and studied by D. Breaz and N. Breaz [14]. (4) For , and , we obtain the integral operator introduced by Selvaraj and Karthikeyan [12]. (5) For , and , we