Schwarz lemma for harmonic mappings in the unit ball

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 $\Vert u{\Vert }_p:={\left({\int }_S|u(\eta ){|}^{p}d\sigma (\eta )\right)}^{1/p}<\infty$, $p\ge 1$ then $|u(x)|\le g_p(|x|)\Vert u{\Vert }_p$ for some smooth sharp function $g_p$ vanishing in $0$. Moreover we provide sharp constant $C_p$ in the inequality $\Vert Du(0)\Vert \le C_p\Vert u{\Vert }_p$. Those two results extend some known result from harmonic mapping theory (\cite[Chapter~VI]{ABR}).

8 pages