Schwarz-Pick Type Estimates for Gradients of Pluriharmonic Mappings of the Unit Ball

Abstract

The main purpose of this paper is to find generalized Schwarz-Pick type inequalities for pluriharmonic functions f of the unit ball $${\mathbb {B}}^n$$ . The estimates for the gradient of |f| and f are given as follows: $$\begin{aligned} \big |\nabla |f|(z)\big |\le \frac{4\sqrt{n}}{\pi (1-|z|)}, \ \ \ z\in {\mathbb {B}}^n \end{aligned}$$ and $$\begin{aligned} |\nabla f(z)|^2\le \frac{2n(1-|f(z)|^2)}{1-|z|}, \ \ \ z\in {\mathbb {B}}^n. \end{aligned}$$ Moreover, by additional assuming the quasiregularity of f, some additional estimates are given in this paper.