On Stable Univalence and Coefficient Estimates for a Class of Pluriharmonic Mappings in Convex Reinhardt Domains

TL;DR

This paper extends classical univalence and coefficient estimate results from harmonic functions to pluriharmonic mappings in several complex variables, establishing new criteria and correspondences in convex Reinhardt domains.

Contribution

It introduces a multidimensional analogue of the Noshiro-Warschawski Theorem, characterizes a class of pluriharmonic mappings, and links their univalence to holomorphic counterparts in higher dimensions.

Findings

Established sufficient conditions for pluriharmonic univalence.

Proved a correspondence between pluriharmonic and holomorphic classes.

Derived sharp coefficient estimates for the class.

Abstract

In this paper, we investigate the geometric properties of complex-valued pluriharmonic mappings defined over convex Reinhardt domains in $\mathbb{C}^n$. We first establish a multidimensional analogue of the Noshiro-Warschawski Theorem, providing sufficient conditions for the univalence of pluriharmonic mappings based on the real part of their partial derivatives. Furthermore, we introduce and study the class $\mathcal{B}_{\mathcal{H}_{n}^{0}}(M)$ of normalized pluriharmonic mappings, characterized by a specific bound on the sum of their second-order partial derivatives. We prove a one-to-one correspondence between this pluriharmonic class and a corresponding class of holomorphic functions, extending known results from the planar harmonic case to higher dimensions. Specifically, we show that a pluriharmonic mapping $f=h+\overline{g}$ is stable pluriharmonic univalent if and only if its holomorphic counterpart $F=h+g$ is stable holomorphic univalent on the unit polydisk $\mathbb{P}\Delta(0;1)$. Finally, we provide sharp coefficient estimates and sufficient conditions for functions to belong to the class $\mathcal{B}_{\mathcal{H}_{n}^{0}}(M)$. Our results generalize several classical theorems in the theory of univalent harmonic functions to the setting of several complex variables.