Fractional differintegral transformations of univalent Meijer's G-functions

Amir Pishkoo,Maslina Darus

doi:10.1186/1029-242x-2012-36

Amir Pishkoo, Maslina Darus

Open Access

https://doi.org/10.1186/1029-242x-2012-36

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Abstract

In this article, the univalent Meijer's G-functions are classified into three types. Certain integral, differential or differintegral transformations preserving the univalence of the Meijer's G-functions, have been discussed. This classification and transformations are based on Kiryakova's studies in representing the generalized hypergeometric functions as fractional differintegral operators of three basic elementary functions. In fact, these transformations are the Erdelyi-Kober operators (m = 1) or their two-tuple compositions (for m = 2) known also as hypergeometric fractional differintegrals. A number of new univalent Meijer's G-functions can be obtained by successive applications of such transformations, being operators of the generalized fractional calculus (GFC). Some new relations are then interpreted for the starlike, convex, and positive real part functions in terms of Meijer's G-functions.

Highlights

One of the main topics in univalent functions theory is dealing with integral or differential operators that are used to obtain new subclasses of univalent functions and their properties
A very general class of such operators have been defined by means of single integrals involving Meijer’s G-functions as kernels, the so-called operators of the generalized fractional calculus (GFC), [2]
It happens that it is enough to use the differintegral operators of the GFC [2], for m = 1,2

Summary

Introduction

One of the main topics in univalent functions theory is dealing with integral or differential operators that are used to obtain new subclasses of univalent functions and their properties. A very general class of such operators have been defined by means of single integrals (or differintegrals) involving Meijer’s G-functions as kernels, the so-called operators of the generalized fractional calculus (GFC), [2].

Results

Conclusion