Schwarz lemma for harmonic mappings in the unit ball

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