Optimal estimates for the gradient of harmonic functions in the unit disk

TL;DR

This paper derives precise bounds for the gradient of harmonic functions within the unit disk, assuming they belong to Hardy spaces, extending previous results related to harmonic mappings and the Bloch constant.

Contribution

It provides sharp constants for gradient estimates of harmonic functions in the unit disk under Hardy space conditions, generalizing prior work by Maz'ya, Kresin, and Colonna.

Findings

Established explicit sharp constants for gradient estimates

Extended previous results to broader Hardy space classes

Connected gradient bounds to the Bloch constant of harmonic mappings

Abstract

Concrete sharp constants in a pointwise estimate of the gradient of a harmonic function in the unit disk are obtained under the assumption that function belong to Hardy space $h^p$, $p\ge 1$. This generalizes some recent result of Maz'ya & Kresin and a result of Colonna related to the Bloch constant of harmonic mappings of the unit disk into itself.