Univalency of convolutions of univalent harmonic right half-plane mappings

TL;DR

This paper investigates the local univalence of convolutions of harmonic right half-plane mappings with specific dilatations, proving local univalence for certain cases and demonstrating non-univalence for others through numerical analysis.

Contribution

It proves local univalence of convolutions for n=1 and shows non-univalence for n≥2, solving an open problem and providing numerical evidence.

Findings

Convolutions are locally univalent for n=1.

Convolutions are not univalent for n≥2.

Numerical computations support theoretical results.

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 $n\geq 2$.