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}).
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