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Bull. Korean Math. Soc. 2018; 55(5): 1419-1431

Online first article March 14, 2018      Printed September 30, 2018

https://doi.org/10.4134/BKMS.b170823

Copyright © The Korean Mathematical Society.

Bi-Lipschitz property and distortion theorems for planar harmonic mappings with $M$-linearly connected holomorphic part

Jie Huang, Jian-Feng Zhu

Huaqiao University, Huaqiao University

Abstract

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Let $f=h+\overline{g}$ be a harmonic mapping of the unit disk $\mathbb D$ with the holomorphic part $h$ satisfying that $h$ is injective and $h(\mathbb D)$ is an $M$-linearly connected domain. In this paper, we obtain the sufficient and necessary conditions for $f$ to be bi-Lipschitz, which is in particular, quasiconformal. Moreover, some distortion theorems are also obtained.

Keywords: harmonic mapping, quasiconformal mapping, bi-Lipschitz mapping, $M$-linearly connected domain

MSC numbers: Primary 30C62; Secondary 30C20, 30F15

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