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Open Access Research

Fractional differintegral transformations of univalent Meijer's G -functions

Amir Pishkoo and Maslina Darus*

Author Affiliations

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600, Selangor Darul Ehsan, Malaysia

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Journal of Inequalities and Applications 2012, 2012:36  doi:10.1186/1029-242X-2012-36

Published: 16 February 2012

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 Erdélyi-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.

Mathematics Subject Classification (2000): 30C45; 33C60; 33E20.

Keywords:
Meijer's G-function; univalent functions; generalized fractional calculus; Erdélyi-Kober operators; starlike functions; convex functions; positive real part functions