Optimal Estimates for the Gradient of Harmonic Functions in the Unit Disk

Let U be the unit disk, p ≥ 1 and let h ^p (U) be the Hardy space of complex harmonic functions. We find the sharp constants C _p and the sharp functions C _p = C _p (z) in the inequality $$|Dw (z)|\leq {C_p}(1-|z|^2)^{-1-1/p} \|w\|_{h^p(\mathbf U)}, w\in h^p(\mathbf U), z\in \mathbf U,$$ in terms of Gaussian hypergeometric and Euler functions. This generalizes some results of Colonna related to the Bloch constant of harmonic mappings of the unit disk into itself and improves some classical inequalities by Macintyre and Rogosinski.