Inclusion Properties of Certain Subclasses of -Valent Functions Associated with the Integral Operator

The purpose of the present paper is to introduce two subclasses of -valent functions by using the integral operator and to investigate various properties for these subclasses. 1. Introduction Let denote the class of functions of the following form: which are analytic and -valent in the open unit disc . Let be the class of functions analytic in satisfying and The class was introduced by Aouf [1] and we note the following:(i)the class was introduced by Padmanabhan and Parvatham [2];(ii)the class was introduced by Pinchuk [3];(iii) is the class of functions with positive real part greater than ;(iv) is the class of functions with positive real part greater than ;(v) is the class of functions with positive real part. From (1), we have if and only if there exists such that It is known that [4] the class is a convex set. Motivated essentially by Jung et al. [5], Liu and Owa [6] introduced the integral operator as follows: For given by (1) and then from (4), we deduce that It is easily verified from (5) that (see [6]) We note that (i) the one-parameter family of integral operator was defined by Jung et al. [5] and studied by Aouf [7] and Gao et al. [8]. (ii) Consider where the operator is the generalized Bernardi-Libera-Livingston integral operator (see [9]). We have the following known subclasses and of the class for , , and which are defined by Next, by using the integral operator , we introduce the following classes of analytic functions for and : We also note that In particular, we set and . The following lemma will be required in our investigation. Lemma 1 (see [10]). Let and and let be a complex-valued function satisfying the following conditions:(i) is continuous in a domain ;(ii) and ;(iii) whenever and . If is analytic in such that and for , then in . Lemma 2 (see [11]). Let be analytic in with and ,？？ . Then, for and , where is given by and this radius is the best possible. Lemma 3 (see [12]). Let be convex and let be starlike in . Then, for analytic in with , is contained in the convex hull of . In this paper, we obtain several inclusion properties of the classes and associated with the operator . 2. Main Results Unless otherwise mentioned, we assume throughout this paper that , , , , and . Theorem 4. One has Proof. We begin by setting where is analytic in with , . Using the identity (6) in (14) and differentiating the resulting equation with respect to , we obtain This implies that We form the functional by choosing and : Clearly, the first two conditions of Lemma 1 are satisfied. Now, we verify condition (iii) as follows: Therefore applying Lemma