Author:
MICHALSKA MAŁGORZATA,MICHALSKI ANDRZEJ M.
Abstract
Clunie and Sheil-Small [‘Harmonic univalent functions’, Ann. Acad. Sci. Fenn. Ser. A. I. Math.9 (1984), 3–25] gave a simple and useful univalence criterion for harmonic functions, usually called the shear construction. However, the application of this theorem is limited to planar harmonic mappings that are convex in the horizontal direction. In this paper, a natural generalisation of the shear construction is given. More precisely, our results are obtained under the hypothesis that the image of a harmonic function is a union of two sets that are convex in the horizontal direction.
Publisher
Cambridge University Press (CUP)
Cited by
2 articles.
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1. A Note on the Horizontal Shears of Harmonic Mappings;Complex Analysis and Operator Theory;2018-05-10
2. A GENERALISATION OF THE CLUNIE–SHEIL-SMALL THEOREM II;Bulletin of the Australian Mathematical Society;2017-03-02