Some properties for $(K,K^{\prime })$-quasiconformal harmonic mappings to half-planes

Let $\mathbb{U}$ be the open unit disk, $\mathrm{\Omega }=\{\omega \in \mathbb{C}:\mathrm{R}\mathrm{e}\omega >a,-\mathrm{\infty }<a<a_0<+\mathrm{\infty }\}$ with $a,a_0\in \mathbb{R}$, and $K\ge 1$ and $K^{\prime }\ge 0$ as two given constants. In this paper, we study some properties of the class $\mathcal{H}(\mathbb{U},\mathrm{\Omega })$ of mappings consisting of those sense-preserving Euclidean harmonic mappings from $\mathbb{U}$ onto $\mathrm{\Omega }$ with the real, normalized conditions of $f(0)=a_0,f_z(0)>0$ and $f_{\overline{z}}(0)$. We first show that a $(K,K^{\prime })$-quasiconformal mapping $f\in \mathcal{H}(\mathbb{U},\mathrm{\Omega })$ need not be Euclidean Lipschitz continuous. Subsequently, we give a sufficient and necessary condition for $f\in \mathcal{H}(\mathbb{U},\mathrm{\Omega })$ to be $(K,K^{\prime })$-quasiconformal. Additionally, coefficient estimates, distortion theorems, and area theorems are obtained.