A generalization of the Clunie--Sheil-Small theorem

TL;DR

This paper extends the shear construction criterion for harmonic functions, allowing for broader applications by considering harmonic mappings whose images are sums of two horizontally convex sets.

Contribution

It introduces a natural generalization of the shear construction, expanding its applicability beyond convex images to sums of two horizontally convex sets.

Findings

Generalized univalence criterion for harmonic mappings.

Broader class of harmonic functions covered.

Potential applications in complex analysis and geometric function theory.

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.