Some sharp Schwarz-Pick type estimates and their applications of harmonic and pluriharmonic functions

TL;DR

This paper establishes sharp Schwarz-Pick type inequalities for harmonic and pluriharmonic functions, providing new bounds and applications such as Lipschitz continuity, higher-order lemmas, and the Bohr phenomenon.

Contribution

It introduces novel sharp estimates and inequalities for harmonic and pluriharmonic functions, extending classical results to Minkowski spaces and various domains.

Findings

Sharp bounds for harmonic functions in Euclidean unit ball

Schwarz-Pick inequalities for pluriharmonic functions in complex domains

Applications to Lipschitz continuity and Bohr phenomenon

Abstract

The purpose of this paper is to study the Schwarz-Pick type inequalities for harmonic or pluriharmonic functions. By analogy with the generalized Khavinson conjecture, we first give some sharp estimates of the norm of harmonic functions from the Euclidean unit ball in $\mathbb{R}^n$ into the unit ball of the real Minkowski space. Next, we give several sharp Schwarz-Pick type inequalities for pluriharmonic functions from the Euclidean unit ball in $\mathbb{C}^n$ or from the unit polydisc in $\mathbb{C}^n$ into the unit ball of the Minkowski space. Furthermore, we establish some sharp coefficient type Schwarz-Pick inequalities for pluriharmonic functions defined in the Minkowski space. Finally, we use the obtained Schwarz-Pick type inequalities to discuss the Lipschitz continuity, the Schwarz-Pick type lemmas of arbitrary order and the Bohr phenomenon of harmonic or pluriharmonic functions.