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

Abstract: In this paper, we investigate the geometric properties of complex-valued pluriharmonic mappings defined over convex Reinhardt domains in $\mathbb{C}<sup>n$. 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 $\mathcal{B}<em>{\mathcal{H}</em>{n}<sup>{0}}(M)$ 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 $\mathbb{P}Δ(0;1)$. Finally, we provide sharp coefficient estimates and sufficient conditions for functions to belong to the class $\mathcal{B}<em>{\mathcal{H}</em>{n}<sup>{0}}(M)$. Our results generalize several classical theorems in the theory of univalent harmonic functions to the setting of several complex variables.