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Applications of differential subordinations for norm estimates of an integral operator

Part of: Geometric function theory

Published online by Cambridge University Press:  06 November 2017

Jacek Dziok ,
Ravinder Krishna Raina  and
Janusz Sokół
Jacek Dziok
Affiliation:
University of Rzeszów, Faculty of Mathematics and Natural Sciences, ul. Prof. Pigonia 1, 35–310 Rzeszów, Poland (jdziok@univ.rzeszow.pl)
Ravinder Krishna Raina
Affiliation:
Maharana Pratap University of Agriculture and Technology, Udaipur, India
Janusz Sokół
Affiliation:
University of Rzeszów, Faculty of Mathematics and Natural Sciences, ul. Prof. Pigonia 1, 35–310 Rzeszów, Poland (jsokol@prz.edu.pl)
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Abstract

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By considering the norm of a locally univalent function given by

we obtain such norm estimates for an operator of functions involving a convolution structure of convex univalent functions with a subclass of convex functions (defined by subordination). We also obtain some inequalities concerning this norm for functions under a certain fractional integral operator. Some implications of our results are briefly pointed out in the concluding section.

Keywords

norm estimatesunivalent functionsstar-like functionssubordinationfractional integral operator

MSC classification

Secondary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.) 30C80: Maximum principle; Schwarz's lemma, Lindelöf principle, analogues and generalizations; subordination
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 148 , Issue 2 , April 2018 , pp. 281 - 291
Copyright
Copyright © Royal Society of Edinburgh 2018