Heinz inequality for the unit ball

TL;DR

This paper generalizes the Schwarz lemma for harmonic mappings of the unit ball, deriving a Heinz inequality with a sharp boundary constant, advancing understanding of harmonic function behavior on the unit ball.

Contribution

It introduces a generalized Schwarz lemma for harmonic mappings and establishes a sharp Heinz inequality on the boundary of the unit ball.

Findings

Proved a generalized Schwarz lemma for harmonic mappings.

Derived a sharp Heinz inequality with optimal constant.

Established boundary behavior bounds for harmonic mappings.

Abstract

We first prove the following generalization of Schwarz lemma for harmonic mappings. Let $u$ be a harmonic mapping of the unit ball onto itself. Then we prove the inequality $\|u(x)-(1-\|x\|^2)/(1+\|x\|^2)^{n/2} u(0)\|\le U(|x| N)$. By using the Schwarz lemma for harmonic mappings we derive Heinz inequality on the boundary of the unit ball by providing a sharp constant $C_n$ in the inequality: $\|\partial_r u(r\eta)\|_{r=1}\ge C_n$, $\|\eta\|=1$, for every harmonic mapping of the unit ball into itself satisfying the condition $u(0)=0$, $\|u(\eta)\|=1$.