The aim of this paper is twofold. First, we obtain a Schwarz–Pick type lemma for the \(\alpha \) -harmonic mapping \(u=P_{\alpha }[\phi ]\) in \(\mathbb {B}^{n}\) , where \(\phi \in L^{p}(\mathbb {S}^{n-1},\mathbb {R} )\) and \(p\in [1,\infty ]\) . We obtain an explicit expression of the function \(g_{n,\alpha ,p} \) in the inequality \(|\nabla u(x)| \le g_{n,\alpha ,p}(|x|)(1-|x|^{2})^{\frac{1-n-p}{p}}\Vert \phi \Vert _{L^p(\mathbb {S}^{n-1}, \mathbb {R} )}\) , which is sharp when \(x=0\) or \(p\in (1,\infty ]\) and \(\alpha =2-n\) or \(p=1\) and \(2-n\le \alpha <1\) . 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, 18:14, 2024) and Khalfallah et al. (Mediterr J Math, 18:19, 2021).
