Convolution Properties of -Valent Functions Associated with a Generalization of the Srivastava-Attiya Operator

Let denote the class of functions analytic in the open unit disc and given by the series . For , the transformation defined by , has been recently studied as fractional differintegral operator by Mishra and Gochhayat (2010). In the present paper, we observed that can also be viewed as a generalization of the Srivastava-Attiya operator. Convolution preserving properties for a class of multivalent analytic functions involving an adaptation of the popular Srivastava-Attiya transform are investigated. 1. Introduction and Preliminaries Let be the class of functions analytic in the open unit disk Suppose that and are in . We say that is subordinate to (or is superordinate to ), written as if there exists a function , satisfying the conditions of the Schwarz lemma and such that It follows that In particular, if is univalent in , then the reverse implication also holds (cf. [1]). For real parameters and such that , recalling the function of the form: which maps conformally onto a disk (whenever ), symmetrical with respect to the real axis, which is centered at the point and with its radius equal to Furthermore, the boundary circle of the disk intersects the real axis at the point and provided . In this paper we will also be dealing with the subclass of consisting of functions of the following form: With a view to define the Srivastava-Attiya transform we recall here a general Hurwitz-Lerch-Zeta function, which is defined in [2, 3] by the following series: Important special cases of the function include, for example, the Reimann zeta function , the Hurwitz zeta function , the Lerch zeta function , the polylogarithm and so on. Recent results on , can be found in the expositions [4, 5]. By making use of the following normalized function: Srivastava-Attiya [2] introduced the linear operator by the following series: where the function is, respectively, by The operator is now popularly known in the literature as the Srivastava-Attiya operator. Various basic properties of are systematically investigated in [6–11]. For a function and represented by the series (8), the transformation defined by has been recently studied as fractional differintegral operator by the authors [12]. We observed that can also be viewed as a generalization of the Srivastava-Attiya operator (take in (14)), suitable for the study of multivalent functions. (Also see [13] for a variant.) Furthermore, transformation generalizes several previously studied familiar operators. For example taking we get the identity transformation; the choices yield the Alexander transformation and a negative integer,