A Generalisation of the Clunie–sheil-Small Theorem

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.

A Generalisation of the Clunie–sheil-Small Theorem Ii

“…In this paper we generalise the theorem of Clunie and Sheil-Small and extend our previous results given in [12]. In Section 2 we prove some topological properties of simply connected sets.…”

“…If in Theorem 4.2 one omits the assumption that both f (D) and (h − g)(D) are simply connected, then the theorem is no longer true (see [12]). Remark 4.3.…”