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arXiv · 1905.10551

On univalent log-harmonic mappings

ZhiHong Liu·Saminathan Ponnusamy
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Abstract

We consider the class univalent log-harmonic mappings on the unit disk. Firstly, we obtain necessary and sufficient conditions for a complex-valued continuous function to be starlike or convex in the unit disk. Then we present a general idea, for example, to construct log-harmonic Koebe mapping, log-harmonic right half-plane mapping and log-harmonic two-slits mapping and then we show precise ranges of these mappings. Moreover, coefficient estimates for univalent log-harmonic starlike mappings are obtained. Growth and distortion theorems for certain special subclass of log-harmonic mappings are studied. Finally, we propose two conjectures, namely, log-harmonic coefficient and log-harmonic covering conjectures.

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ZhiHong Liu, Saminathan Ponnusamy. 2019-05-25. On univalent log-harmonic mappings. https://arxiv.org/abs/1905.10551

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