Marković: Optimal estimates for the gradient of harmonic functions in the unit disk, Complex analysis and operator theory: DOI

Abstract. Let U be the unit disk, p> 1 and let hp(U) be the Hardy space of complex harmonic functions. We find the sharp constants Cp and the sharp functions Cp = Cp(z) in the inequality |Dw(z) | ≤ Cp(1 − |z|2)−1−1/p‖w‖hp(U), w ∈ hp(U), z ∈ U, in terms of Gauss hypergeometric and Euler functions. This generalizes some results of Colonna related to the Bloch constant of harmonic map-pings of the unit disk into itself and improves some classical inequalities by Macintyre and Rogosinski