Univalence criterion for harmonic mappings and $\Phi$-like functions
Abstract
In this paper, we obtain a new characterization for univalent harmonic mappings and obtain a structural formula for the associated function which defines the analytic -like functions in the unit disk. The new criterion stated in this article for the injectivity of harmonic mappings implies the well-known results of Kas'yanyuk \cite{Kas59} and Brickman \cite{Brick73} for analytic functions, but with a simpler proof than theirs. A number of consequences of the characterization, and examples are also presented. Further investigation provides a new method to construct univalent harmonic mappings with the help of an improved distortion theorem.
Cite
@article{arxiv.1510.04886,
title = {Univalence criterion for harmonic mappings and $\Phi$-like functions},
author = {Sergey Yu. Graf and Saminathan Ponnusamy and Victor V. Starkov},
journal= {arXiv preprint arXiv:1510.04886},
year = {2016}
}
Comments
11 pages, 1 figure and the article is with some journal for publication; Shorter version of an earlier version