Some properties for (K,K′)-quasiconformal harmonic mappings to half-planes

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Abstract

Let $\mathbb{U}$ be the open unit disk, $\Omega=\left\{\omega \in \mathbb{C}: \operatorname{Re} \omega>a,-\infty<a<a_{0}<+\infty\right\}$ with $a, a_{0} \in \mathbb{R}$, and K ≥ 1 and K^′ ≥ 0 as two given constants. In this paper, we study some properties of the class $\mathcal{H}(\mathbb{U}, \Omega)$ of mappings consisting of those sense-preserving Euclidean harmonic mappings from $\mathbb{U}$ onto Ω with the real, normalized conditions of $f(0)=a_{0}, f_{z}(0)>0$ and $f_{\bar{z}}(0)$. We first show that a (K,K^′)- quasiconformal mapping $f \in \mathcal{H}(\mathbb{U}, \Omega)$ need not be Euclidean Lipschitz continuous. Subsequently, we give a sufficient and necessary condition for $f \in \mathcal{H}(\mathbb{U}, \Omega)$ to be (K,K^′)-quasiconformal. Additionally, coefficient estimates, distortion theorems, and area theorems are obtained.

Keywords

harmonic mapping / (K,K^′)-quasiconformal mapping / distortion theorem / coefficient estimate / area theorem

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