If 1⩽p⩽∞\documentclass[12pt]{minimal}
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\begin{document}$$1\leqslant p\leqslant \infty $$\end{document} and u is a harmonic mapping of the unit ball Bn\documentclass[12pt]{minimal}
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\begin{document}$$B^n$$\end{document} into itself such that u(0)=0\documentclass[12pt]{minimal}
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\begin{document}$$u(0) = 0$$\end{document}, then we obtain the sharp inequality |u(x)|⩽gp(|x|)‖u‖p\documentclass[12pt]{minimal}
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\begin{document}$$|u(x)|\leqslant g_p(|x|)\Vert u\Vert _p$$\end{document} for some smooth function gp\documentclass[12pt]{minimal}
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\begin{document}$$g_p$$\end{document} vanishing in 0, where ‖u‖∞:=supx∈Bn|u(x)|\documentclass[12pt]{minimal}
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\begin{document}$$\Vert u\Vert _\infty := \sup _{x\in B^n}|u(x)|$$\end{document} for p=+∞\documentclass[12pt]{minimal}
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\begin{document}$$p = +\infty $$\end{document}, and otherwise ‖u‖p:=supr∫S|u(rη)|pdσ(η)1/p.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Vert u\Vert _p:=\sup _r\left( \int _S|u(r\eta )|^pd\sigma (\eta )\right) ^{1/p}. \end{aligned}$$\end{document}Here σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document} is the normalized rotationally invariant Borel measure on the unit sphere S:=∂Bn\documentclass[12pt]{minimal}
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\begin{document}$$S := \partial B^n$$\end{document}. Moreover, we obtain an explicit form of the sharp constant Cp\documentclass[12pt]{minimal}
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\begin{document}$$C_p$$\end{document} in the inequality ‖Du(0)‖⩽Cp‖u‖p\documentclass[12pt]{minimal}
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\begin{document}$$\Vert Du(0)\Vert \leqslant C_p\Vert u\Vert _p$$\end{document}. Those two results extend some known result from harmonic mapping theory (Axler et al. in Harmonic function theory, Springer, New York, 1992, Chapter VI).
