On univalent log-harmonic mappings
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.
Cite
@article{arxiv.1905.10551,
title = {On univalent log-harmonic mappings},
author = {ZhiHong Liu and Saminathan Ponnusamy},
journal= {arXiv preprint arXiv:1905.10551},
year = {2019}
}
Comments
16 pages; This paper was with Studia Scientiarum Mathematicarum Hungarica since May 2017; Finally returned by saying that they could not find a suitable referee!