A generalization of the Clunie--Sheil-Small theorem
Complex Variables
2014-09-29 v1
Abstract
In 1984, a simple and useful univalence criterion for harmonic functions was given by Clunie and Sheil-Small, which is usually called the shear construction. However, the application of this theorem is limited to the planar harmonic mappings convex in the horizontal direction. In this paper, a natural generalization of the shear construction is given. More precisely, our results are obtained under the hypothesis that the image of a harmonic mapping is a sum of two sets convex in the horizontal direction.
Keywords
Cite
@article{arxiv.1409.7496,
title = {A generalization of the Clunie--Sheil-Small theorem},
author = {Małgorzata Michalska and Andrzej Michalski},
journal= {arXiv preprint arXiv:1409.7496},
year = {2014}
}