Univalency of convolutions of univalent harmonic right half-plane mappings

Abstract

We consider the convolution of half-plane harmonic mappings with respective dilatations $(z+a)/(1+az)$ and $e^{i\theta }{z}^n$, where $-1<a<1$ and $\theta \in \mathbb{R},n\in \mathbb{N}$. We prove that such convolutions are locally univalent for $n=1$, which solves an open problem of Dorff et. al (see \cite[Problem~3.26]{Bshouty2010}). Moreover, we provide some numerical computations to illustrate that such convolutions are not univalent for $n\ge 2$.

11 pages, 8 figures; Will appear in the Journal "Computational Methods and Function Theory"