Weighted Lipschitz continuity, Schwarz-Pick's Lemma and Landau-Bloch's theorem for hyperbolic-harmonic mappings in $\mathbb{C}^{n}$

TL;DR

This paper explores properties of hyperbolic-harmonic mappings in complex n-dimensional space, establishing key theorems and constants related to Lipschitz continuity, Schwarz-Pick lemma, and Landau-Bloch theorem.

Contribution

It introduces new relationships between weighted Lipschitz functions and hyperbolic-harmonic Bloch spaces, and proves a Schwarz-Pick type theorem with applications to Landau-Bloch constants.

Findings

Established the relationship between weighted Lipschitz functions and hyperbolic-harmonic Bloch spaces.

Proved a Schwarz-Pick type theorem for hyperbolic-harmonic mappings.

Demonstrated the existence of Landau-Bloch constants for mappings in α-Bloch spaces.

Abstract

In this paper, we discuss some properties on hyperbolic-harmonic mappings in the unit ball of $\mathbb{C}^{n}$. First, we investigate the relationship between the weighted Lipschitz functions and the hyperbolic-harmonic Bloch spaces. Then we establish the Schwarz-Pick type theorem for hyperbolic-harmonic mappings and apply it to prove the existence of Landau-Bloch constant for mappings in $\alpha$-Bloch spaces.