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arXiv · 1506.06410

Schwarz lemma for harmonic mappings in the unit ball

David Kalaj
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Abstract

We prove the following generalization of Schwarz lemma for harmonic mappings. If $u$ is a harmonic mapping of the unit ball $B_n$ onto itself such that $u(0)=0$ and $\|u\|_p:=\left(\int_S|u(η)|^pdσ(η)\right)^{1/p}<\infty$, $p\ge 1$ then $|u(x)|\le g_p(|x|)\|u\|_p$ for some smooth sharp function $g_p$ vanishing in $0$. Moreover we provide sharp constant $C_p$ in the inequality $\|Du(0)\|\le C_p\|u\|_p$. Those two results extend some known result from harmonic mapping theory (\cite[Chapter~VI]{ABR}).

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David Kalaj. 2015-06-21. Schwarz lemma for harmonic mappings in the unit ball. https://arxiv.org/abs/1506.06410

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