Schwarz lemma for hyperbolic harmonic mappings in the unit ball

Assume that $p\in[1,\infty]$ and $u=P_{h}[\phi]$, where $\phi\in L^{p}(\mathbb{S}^{n-1},\mathbb{R}^n)$ and $u(0) = 0$. Then we obtain the sharp inequality $|u(x)|\le G_p(|x|)\|\phi\|_{L^{p}}$ for some smooth function $G_p$ vanishing at $0$. Moreover, we obtain an explicit form of the sharp constant $C_p$ in the inequality $\|Du(0)\|\le C_p\|\phi\|_{L^{p}}$. These two results generalize and extend some known result from harmonic mapping theory (\cite[Theorem 2.1]{kalaj2018}) and hyperbolic harmonic theory (\cite[Theorem 1]{bur}).