A unified framework for Bohr-type inequalities using multiple Schwarz functions

TL;DR

The paper addresses Bohr-type inequalities for $K$-quasiconformal harmonic mappings in the unit disk by introducing a unified framework that employs multiple Schwarz functions $\omega_p, \,\omega_m, \,\omega_q$. The authors derive sharp multi-Schwarz Bohr-type bounds, with radii $r_{p,m,k}$ governed by root equations such as $G_{p,m,k}(r)=0$, and show these results recover classical Bohr phenomena (e.g., Theorem C) as special cases while simultaneously strengthening the bounds through derivative and area terms. The main contributions include a sharp, general radius formula, several refined Bohr-type inequalities, and a demonstration that the framework unifies and extends existing results; special cases recover classical bounds and the Bohr radius emerges as a limiting instance. The work provides a versatile toolkit for Bohr phenomena in harmonic mappings and suggests promising avenues for extensions to starlike/convex, pluriharmonic, and Bohr-Rogosinski settings, with supportive numerical tables and figures illustrating radius behavior.

Abstract

This paper introduces a unified framework for Bohr-type inequalities by incorporating multiple Schwarz functions into the majorant series for $K$-quasiconformal harmonic mappings in the unit disk $\mathbb{D} := \{z\in\mathbb{C} : |z| < 1\}$. In this study, we establish several improved and refined versions of the Bohr inequality that generalize and interconnect numerous known results. Our approach not only systematically recovers the existing theorems as special cases but also generates new results that are inaccessible through single-function methods.

Figures (3)

• Figure 1: The position of $r_{p,m,k}$, the smallest root of equation (\ref{['p3']}) in $(0, 1)$, varies with different values of $p, m$, and $k$
• Figure 2: The position of $r_{m,p,q,k}$ in $(0,1)$ for different values of $m,p,q\in\mathbb{N}$ and $k\in[0, 1)$
• Figure 3: The position of $r_{m,p,q,k}$ in $(0,1)$ for different values of $m, p,q\in\mathbb{N}$ and $k\in[0,1)$

Theorems & Definitions (18)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Theorem 3.1
• proof
• Remark 3.1
• Remark 3.2
• Theorem 3.2