Schwarz lemma for hyperbolic harmonic mappings in the unit ball

TL;DR

This paper establishes sharp inequalities for hyperbolic harmonic mappings in the unit ball, extending classical results by providing explicit bounds and constants related to the mappings' derivatives and norms.

Contribution

It generalizes known harmonic and hyperbolic harmonic results by deriving sharp inequalities and explicit constants for hyperbolic harmonic mappings in the unit ball.

Findings

Derived a sharp inequality |u(x)| ≤ G_p(|x|)‖φ‖_{L^p} with a smooth function G_p

Obtained an explicit form of the sharp constant C_p in the derivative inequality

Extended classical harmonic mapping results to hyperbolic harmonic mappings

Abstract

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}).