Geometric Properties of Normalized Wright Functions

Abstract

In this chapter, we investigate some subclasses of analytic functions in the open unit disk in the complex plane. We derive characteristic properties of the normalized Wright functions belonging to these classes and we find upper bound estimate for these functions belonging to the subclasses studied. Several sufficient conditions were obtained for the parameters of the normalized form of the Wright functions to be in this class. Some geometric properties of the integral transforms represented with the normalized Wright functions are also studied. We give some sufficient conditions for the integral operators involving normalized Wright functions to be univalent in the open unit disk. The key tools in our proofs are the Becker’s and the generalized version of the well-known Ahlfor’s and Becker’s univalence criteria. In the final section, we introduce a Poisson distribution series, whose construction is alike Wright functions, and obtain necessary and sufficient conditions for this series belonging to the class \(S^{*} C(\alpha ,\beta ;\gamma )\), and necessary and sufficient conditions for those belonging to the class \(TS^{*} C(\alpha ,\beta ;\gamma )\). We also introduce two integral operators related to this series and investigate various geometric properties of these integral operators.