Differential Operators, Multiple Schwarz Functions, and the Bohr Radius of Stable Harmonic Maps

TL;DR

This work advances the Bohr radius theory for stable harmonic mappings by studying Bohr-type inequalities associated with differential operators $D f = z f_{z} - \overline{z} f_{\bar{z}}$ and $\mathscr{D} f = z f_{z} + \overline{z} f_{\bar{z}}$ and by incorporating multiple Schwarz functions $\omega_m \in \mathcal{B}_m$. It derives sharp Bohr and Bohr-Rogosinski inequalities for both univalent and convex stable harmonic classes, with explicit radii $r_{1,m,p}, r_{2,m,p}, r_{3,s,m,p,q}, r_{4,s,m,p,q}$ (and variants for the second-order operators), showing best-possible constants via Koebe-type extremals. The paper further extends these results to second-order harmonic differential operators, yielding radii $r_{5,m}, r_{6,m}, r_{7,m,p}, r_{8,m,p}$ for linear combinations with $D$ and $\ ext{\mathscr{D}}$, and proves sharpness with concrete extremal examples. Finally, it proves an improved Bohr inequality in terms of $S_r/\pi$ for stable harmonic mappings, expressing $S_r/\pi$ via Parseval and obtaining a sharp radius $r_{m,k}$ for a polynomial $P_k$, thus answering open questions about Bohr phenomena in this setting and strengthening connections to classical results such as Xu-An-Liu-JA-2025.

Abstract

In this paper, we study the Bohr phenomenon for differential operators $D$ and $\mathscr{D}$ of stable harmonic mappings involving multiple Schwarz functions in $\mathcal{B}_n$, using distance formulations. By constructing suitable combinations of multiple Schwarz functions, we establish sharp and improved Bohr-type inequalities for these mappings. The corresponding Bohr radii are also determined for certain subclasses of stable harmonic functions and their associated differential operators. Moreover, Bohr-Rogosinski-type inequalities are derived, which highlights the influence of multiple Schwarz functions on the geometric properties of stable harmonic mappings. All the radii are determined, and we prove that each one is the best possible.

Figures (4)

• Figure 1: The graphs exhibit the locations of the roots $r_{1,m,p}$ in $(0,1)$ for different values of $m,p$.
• Figure 2: The graphs exhibit the locations of the roots $r_{2,m,p}$ in $(0,1)$ for different values of $m,p$.
• Figure 3: The graphs exhibit the locations of the roots $r_{3,s,m,p,q}$ in $(0,1)$ for different values of $s,m,p,q$.
• Figure 4: The graphs exhibit the locations of the roots $r_{4,s,m,p,q}$ in $(0,1)$ for different values of $s,m,p,q$.

Theorems & Definitions (32)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Theorem 3.1
• Remark 3.1
• proof : Proof of Theorem \ref{['Th-3.1']}
• Remark 3.2