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International Journal of Mathematics and Mathematical Sciences
Volume 2017 (2017), Article ID 4015268, 4 pages
https://doi.org/10.1155/2017/4015268
Research Article

Convolutions of Harmonic Functions with Certain Dilatations

Mathematical Sciences, Kent State University, Burton, OH 44021-9500, USA

Correspondence should be addressed to Jay M. Jahangiri; ude.tnek@ignahajj

Received 1 October 2017; Accepted 13 November 2017; Published 29 November 2017

Academic Editor: Teodor Bulboaca

Copyright © 2017 Om P. Ahuja and Jay M. Jahangiri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The convolution of harmonic functions, unlike the analytic case, proved to be very challenging. In this paper, we introduce dilatation conditions that guarantee the convolution of two harmonic functions to be locally one-to-one, sense-preserving, and close-to-convex harmonic in the unit disk.