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

TL;DR

This paper investigates the Bohr phenomenon for differential operators acting on stable harmonic mappings involving multiple Schwarz functions, establishing sharp inequalities and determining optimal Bohr radii for various subclasses.

Contribution

It introduces new sharp Bohr-type inequalities for stable harmonic maps with multiple Schwarz functions and identifies the best possible Bohr radii for these classes.

Findings

Established sharp Bohr inequalities for differential operators on harmonic mappings.

Determined the exact Bohr radii for specific subclasses of stable harmonic functions.

Derived Bohr-Rogosinski inequalities highlighting the role of multiple Schwarz functions.

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.