Schwarz lemma for hyperbolic harmonic mappings in the unit ball

Abstract

Assume that $p\in [1,\infty ]$ and $u=P_h[\varphi ]$, where $\varphi \in L^p({\mathbb{S}}^{n-1},{\mathbb{R}}^n)$ and $u(0)=0$. Then we obtain the sharp inequality $|u(x)|\le G_p(|x|)\Vert \varphi {\Vert }_{L^p}$ for some smooth function $G_p$ vanishing at $0$. Moreover, we obtain an explicit form of the sharp constant $C_p$ in the inequality $\Vert Du(0)\Vert \le C_p\Vert \varphi {\Vert }_{L^p}$. 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}).

20 pages

Cited by in corpus (1)

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