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A GENERALISATION OF THE CLUNIE–SHEIL-SMALL THEOREM

Part of: Geometric function theory Two-dimensional theory

Published online by Cambridge University Press:  17 February 2016

MAŁGORZATA MICHALSKA  and
ANDRZEJ M. MICHALSKI
MAŁGORZATA MICHALSKA
Affiliation:
Institute of Mathematics, Maria Curie-Skłodowska University, pl. M. Curie-Skłodowskiej 1, 20-031 Lublin, Poland email malgorzata.michalska@poczta.umcs.lublin.pl
ANDRZEJ M. MICHALSKI*
Affiliation:
Department of Complex Analysis, The John Paul II Catholic University of Lublin, ul. Konstantynów 1H, 20-708 Lublin, Poland email amichal@kul.lublin.pl
*
✉amichal@kul.lublin.pl
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Abstract

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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.

Keywords

harmonic mappingsconvex in one directionshear construction

MSC classification

Primary: 31A05: Harmonic, subharmonic, superharmonic functions
Secondary: 30C55: General theory of univalent and multivalent functions 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 94 , Issue 1 , August 2016 , pp. 92 - 100
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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