Annales Academić Scientiarum Fennicć
Mathematica
Volumen 37, 2012, 539-556

A SIMPLE DEFORMATION OF QUASICONFORMAL HARMONIC MAPPINGS IN THE UNIT DISK

Dariusz Partyka and Ken-ichi Sakan

The John Paul II Catholic University of Lublin, Faculty of Mathematics and Natural Sciences
Al. Raclawickie 14, P.O. Box 129, 20-950 Lublin, Poland, and
The State School of Higher Education in Chelm, Institute of Mathematics and Information Technology
Pocztowa 54, 22-100 Chelm, Poland; partyka 'at' kul.lublin.pl

Osaka City University, Graduate School of Science, Department of Mathematics
Sugimoto, Sumiyoshi-ku, Osaka, 558, Japan; ksakan 'at' sci.osaka-cu.ac.jp

Abstract. Given a sense-preserving injective harmonic mapping F in the unit disk D and a \in C we consider a simple deformation C \ni a \mapsto F_a := H + a \overline G of F, where H and G are holomorphic mappings in D determined by F = H + \overline G and G(0) = 0. We introduce a natural generalization of convexity called alpha-convexity. Then we study the bi-Lipschitz behaviour of mappings F_a under the assumption that F is a quasiconformal harmonic mapping of D onto an alpha-convex domain F(D). As an application we show that if F is a quasiconformal harmonic self-mapping of D, then H is a bi-Lipschitz mapping. Consequently, a sense-preserving harmonic self-mapping F of D is quasiconformal iff H is Lipschitz with the Jacobian of F separated from zero by a positive constant in D.

2010 Mathematics Subject Classification: Primary 30C55, 30C62.

Key words: Harmonic mappings, Lipschitz condition, bi-Lipchitz condition, co-Lipchitz condition, Jacobian, quasiconformal mappings.

Reference to this article: D. Partyka and K. Sakan: A simple deformation of quasiconformal harmonic mappings in the unit disk. Ann. Acad. Sci. Fenn. Math. 37 (2012), 539-556.

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doi:10.5186/aasfm.2012.3731