On Stable Univalence and Coefficient Estimates for a Class of Pluriharmonic Mappings in Convex Reinhardt Domains
Molla Basir Ahamed, Sujoy Majumder, Debabrata Pramanik
TL;DR
This work develops a multidimensional Noshiro–Warschawski–type univalence criterion for pluriharmonic mappings on convex Reinhardt domains, showing that if $\mathrm{Re}(e^{i\gamma}\frac{\partial h}{\partial z_j})>|\frac{\partial g}{\partial z_j}|$ for all $j$, then $f=h+\overline{g}$ is univalent. It then proves that stable pluriharmonic univalence of $f$ on the unit polydisk is equivalent to stable holomorphic univalence of $F=h+g$ on the same domain and establishes a one‑to‑one correspondence between $\mathcal{B}_{\mathcal{H}_n^0}(M)$ and $\mathcal{B}_n(M)$ via $F_{\varepsilon}=h+\varepsilon g$ for all $|\varepsilon|=1$. The paper also derives sharp multidimensional coefficient bounds $\sum_{|\alpha|=m}|a_{\alpha}|\leq \frac{\binom{m+n-1}{n-1}M}{nm(m-1)}$ and a parallel bound for $|b_{\alpha}|$, gives sufficient conditions for membership in $\mathcal{B}_{\mathcal{H}_n^0}(M)$, proves closure under convex combinations, and establishes growth estimates $n\|z\|_{\infty}-\frac{M n^2}{2}\|z\|_{\infty}^2\leq |f(z)|\leq n\|z\|_{\infty}+\frac{M n^2}{2}\|z\|_{\infty}^2$, all with sharpness.
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 $\mathbb{P}Δ(0;1)$. 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.