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 Bi-Lipschitz property and distortion theorems for planar harmonic mappings with $M$-linearly connected holomorphic part Bull. Korean Math. Soc. 2018 Vol. 55, No. 5, 1419-1431 https://doi.org/10.4134/BKMS.b170823Published online September 30, 2018 Jie Huang, Jian-Feng Zhu Huaqiao University, Huaqiao University Abstract : 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 Full-Text :