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

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 $\mathbb{P}Δ(0;1)$. 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.

💡 Research Summary

The paper investigates pluriharmonic mappings defined on convex Reinhardt domains in ℂⁿ, extending several classical results from planar harmonic function theory to the several‑complex‑variables setting. After recalling basic notions such as Reinhardt domains, starlikeness, convexity, and close‑to‑convexity, the authors establish a multidimensional analogue of the Noshiro‑Warschawski theorem (Proposition 1.1). They show that if all first‑order partial derivatives ∂f/∂z_j are non‑zero at a point z₀ and continuous, then f is locally univalent in a small convex Reinhardt neighbourhood of z₀. The proof exploits the rotational invariance of Reinhardt domains, constructs a linear interpolation path between two points, and uses a mean‑value argument to obtain a positive lower bound for the difference f(w)−f(z).

Proposition 1.2 provides the canonical decomposition f = h + g for pluriharmonic functions on simply connected domains, where both h and g are holomorphic. This is the higher‑dimensional counterpart of the classical Clunie‑Sheil‑Small representation and serves as the foundation for the subsequent analysis.

The central contribution is the introduction of the class 𝔅_{ℋ_n⁰}(M). A normalized pluriharmonic mapping f = h + g belongs to this class if, for every z in the unit polydisk 𝔓Δ(0;1),

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