Optimal estimates for harmonic functions in the unit ball

Abstract: We find the sharp constants $C_p$ and the sharp functions $C_p=C_p(x)$ in the inequality $$|u(x)|\leq \frac{C_p}{(1-|x|^2)^{(n-1)/p}}|u|_{h^p(B^n)}, u\in h^p(B^n), x\in B^n,$$ in terms of Gauss hypergeometric and Euler functions. This extends and improves some results of Axler, Bourdon and Ramey (\cite{ABR}), where they obtained similar results which are sharp only in the cases $p=2$ and $p=1$.