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