Inclusion Properties of New Classes of Analytic Functions

The purpose of the present paper is to introduce certain new subclasses of analytic functions defined by Srivastava-Attiya operator and study their inclusion relationships and to obtain some interesting consequences of the inclusion relations. 1. Introduction, Definitions, and Preliminaries Let be the class of analytic functions defined on the unit disk , normalized by the conditions . Let be the subclass of consisting of univalent functions. Let denote the class of analytic convex univalent functions defined in satisfying the conditions and for all . For any two analytic functions and defined in the unit disc , we say that is subordinate to in , denoted by , if is univalent in , and . The convolution or the Hadamard product of two analytic functions with power series representations convergent in is the function , with power series representation The following classes of functions were defined by Shanmugam [1]. For a fixed analytic function and , We note that . When and , , the class of starlike functions; , the class of convex functions; and , the class of close to convex functions. The generalized Hurwitz-Lerch Zeta function is defined as where？？ . This function contains, as its special cases, functions such as the Riemann and Hurwitz zeta function, Lerch zeta function, the polylogarithmic function, and the Lipschitz Lerch zeta function. Several interesting properties and characteristics of the Hurwitz-Lerch zeta function can be found in the recent investigations by Choi and Srivastava [2], Ferreira and López [3], Garg et al. [4], Lin and Srivastava [5], and Lin et al. [6]. Using this function, Srivastava and Attiya [7] introduced the following family of linear operator , defined by where the function is given by Using (6) in (5), we get For and , where denotes Alexander transform [8] and denotes Bernardi integral operator [9]. By (7), Srivastava and Attiya obtained the following relation: In order to obtain our main results, we need the following lemmas. Lemma 1 (see [10]). Let and be constants. Let be convex univalent in with and for in . Let be analytic in with . Then Lemma 2 (see [11]). Let and be constants. Let be convex univalent in with and for in . Let be analytic in with . Let be analytic in with . Then Lemma 3 (see [12]). Let a function be analytic in and . If there exists a point such that and , then where and . 2. Main Results First we will define a subclass as follows. Definition 4. For a fixed analytic function and a real , Remark 5. When , , and , reduces to the class . Theorem 6. For ,？？ . Proof. Let and . Note that is analytic and .