2014/09/26 by Małgorzata Michalska, Michalska, Małgorzata, Andrzej Michalski +1
Mathematics · #Algebraic and Geometric Analysis #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.1409.7496
openalex publication_date 2014/09/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.