Convolutions of slanted half-plane harmonic mappings

Let ${\mathcal S^0}(H_{\gamma})$ denote the class of all univalent, harmonic, sense-preserving and normalized mappings $f$ of the unit disk \ID onto the slanted half-plane $H_\gamma :=\{w:\,{\rm Re\,}(e^{i\gamma}w) >-1/2\}$ with an additional condition $f_{\bar{z}}(0)=0$. Functions in this class can be constructed by the shear construction due to Clunie and Sheil-Small which allows by examining their conformal counterpart. Unlike the conformal case, convolution of two univalent harmonic convex mappings in \ID is not necessarily even univalent in \ID. In this paper, we fix $f_0\in{\mathcal S^0}(H_{0})$ and show that the convolutions of $f_0$ and some slanted half-plane harmonic mapping are still convex in a particular direction. The results of the paper enhance the interest among harmonic mappings and, in particular, solves an open problem of Dorff, et. al. \cite{DN} in a more general setting. Finally, we present some basic examples of functions and their corresponding convolution functions with specified dilatations, and illustrate them graphically with the help of MATHEMATICA software. These examples explain the behaviour of the image domains.Comment: 15 pages, preprint of December 2011 (submitted to a journal for publication

Note on the convolution of harmonic mappings

Dorff et al. \cite{DN} formulated a question concerning the convolution of two right half-plane mappings, where the normalization of the functions was considered incorrectly. In this paper, we have reformulated the open problem in correct form and provided a solution to it in a more general form. In addition, we also obtain two new theorems which correct and improve some other results.Comment: 11 pages; An extended version of this article was in a couple of conferences, and also in later workshops in Chennai during 2017 in India. This version will appear in Bulletin of the Australian Mathematical Societ