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A GENERALISATION OF THE CLUNIE–SHEIL-SMALL THEOREM II

Part of: Geometric function theory Two-dimensional theory

Published online by Cambridge University Press:  02 March 2017

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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We study properties of the simply connected sets in the complex plane, which are finite unions of domains convex in the horizontal direction. These considerations allow us to state new univalence criteria for complex-valued local homeomorphisms. In particular, we apply our results to planar harmonic mappings obtaining generalisations of the shear construction theorem due to Clunie and Sheil-Small [‘Harmonic univalent functions’, Ann. Acad. Sci. Fenn. Ser. A. I. Math.9 (1984), 3–25].

Keywords

harmonic mappingsimply connected setconvexity in the horizontal direction

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 95 , Issue 3 , June 2017 , pp. 457 - 466
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

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