Heinz inequality for the unit ball

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 $\Vert u(x)-(1-\Vert x{\Vert }^2)/(1+\Vert x{\Vert }^2{)}^{n/2}u(0)\Vert \le U(|x|N)$. 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: $\Vert {\partial }_{r}u(r\eta ){\Vert }_{r=1}\ge C_n$, $\Vert \eta \Vert =1$, for every harmonic mapping of the unit ball into itself satisfying the condition $u(0)=0$, $\Vert u(\eta )\Vert =1$.

6 pages

References in corpus (1)

• Some properties of a class of elliptic partial differential operators