Bohr inequality and Bohr-Rogosinski inequality for $K$-Quasiconformal harmonic mappings

TL;DR

This paper establishes sharp Bohr and Bohr-Rogosinski inequalities for certain $K$-quasiconformal harmonic mappings, extending known results to broader classes of functions with specific subordination properties.

Contribution

It introduces new sharp inequalities for $K$-quasiconformal harmonic mappings with analytic parts subordinate to concave, convex, and starlike functions, generalizing previous findings.

Findings

Sharp Bohr-type inequalities for concave univalent functions.

Bohr-Rogosinski inequalities for convex and starlike subordinate functions.

Generalization of existing inequalities to broader classes of harmonic mappings.

Abstract

In this paper, we prove several sharp Bohr-type and Bohr-Rogosinski-type inequalities for $K$-quasiconformal, sense-preserving harmonic mappings on $\mathbb{D}$, whose analytic part is subordinate to a function belonging to the class of concave univalent functions on $\mathbb{D}$. In addition, we derive Bohr-type inequalities for $K$-quasiconformal, sense-preserving harmonic mappings on $\mathbb{D}$, where the analytic part is subordinate to a function from the Ma-Minda class of convex and starlike functions. The results generalize several existing results.