Schwarz-Pick type lemma and Landau type theorem for $\alpha$-harmonic mappings

Abstract

The aim of this paper is twofold. First, we obtain a Schwarz-Pick type lemma for the $\alpha$-harmonic mapping $u=P_{\alpha }[\varphi ]$, where $\varphi \in L^p({\mathbb{S}}^{n-1},\mathbb{R})$ and $p\in [1,\infty ]$. We get an explicit form of the sharp function ${\mathbf{C}}_{\alpha ,q}(x)$ in the inequality $|\nabla u(x)|\le {\mathbf{C}}_{\alpha ,q}(x)\Vert \varphi {\Vert }_{L^p({\mathbb{S}}^{n-1},\mathbb{R})}$. Second, we prove a Landau type theorem for $u=P_{\alpha }[\varphi ]$, where $\varphi \in L^{\infty }({\mathbb{S}}^{n-1},{\mathbb{R}}^n)$. These results generalize and extend the corresponding results due to Kalaj (Complex Anal. Oper. Theory, 2024) and Khalfallah et al. (Mediterr. J. Math., 2021).