Integral Transforms of Functions to Be in a Class of Analytic Functions Using Duality Techniques

Let , denote the class of all normalized analytic functions in the unit disc such that for some with , and . Let , , denote the Pascu class of -convex functions given by the analytic condition which unifies the classes of starlike and convex functions. The aim of this paper is to find conditions on so that the integral transform of the form carry functions from into . As for the applications, for specific values of , it is found that several known integral operators carry functions from into . The results for a more generalized operator related to are also given. 1. Introduction Let denote the class of all functions analytic in the open unit disc with the normalization , and let be the class of functions that are univalent in . A function is said to be starlike or convex , if maps conformally onto the domains, respectively, starlike with respect to origin and convex. Note that in , if follows from the well-known Alexander theorem (see [1] for details). A useful generalization of the class is the class that has the analytic characterization and . Various generalizations of classes and are abundant in the literature. One such generalization is the following. A function is said to be in the Pascu class of -convex functions of order if [2] or in other words This class is denoted by . Even though this class is known as Pascu class of -convex functions of order , since we use the parameter for another important class, we denote this class by , , and we remark that, in the sequel, we only consider the class . Clearly, and , which implies that this class is a smooth passage between the classes of starlike and convex functions. The main objective of this work is to find conditions on the nonnegative real valued integrable function satisfying , such that the operator is in the class . Note that this operator was introduced in [3]. To investigate this admissibility property, the class to which the function belongs is important. Let , where？？ , , and , denote the class of all normalized analytic functions in the open unit disc such that for some . This class and its particular cases were considered by many authors so that the corresponding operator given by (3) is univalent and in for some particular values of , , , and . This work was motivated in [3] by studying the conditions under which and was generalized in [4] by studying the case . Similar situation for the convex case, namely, , was initiated in [5]. After several generalizations by many authors, recently, the conditions under which were obtained in [6] and the corresponding results for the convex case so