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arXiv · 1609.05282

Univalency of convolutions of univalent harmonic right half-plane mappings

Zhihong Liu·Saminathan Ponnusamy
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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 $n\geq 2$.

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Zhihong Liu, Saminathan Ponnusamy. 2016-09-17. Univalency of convolutions of univalent harmonic right half-plane mappings. https://arxiv.org/abs/1609.05282

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