The Bohr's Phenomenon for the class of K-quasiconformal harmonic mappings

TL;DR

This paper establishes sharp versions of Bohr inequalities for K-quasiconformal harmonic mappings in the unit disk, introducing new bounds and radii using advanced coefficient and function analysis techniques.

Contribution

It provides the first sharp Bohr-type inequalities and radii specifically for K-quasiconformal harmonic mappings, refining existing bounds and methods.

Findings

Derived sharp Bohr inequalities for K-quasiconformal harmonic mappings.

Established the sharp Bohr-Rogosinski radius under half-plane conditions.

Introduced new techniques using the quantity S_ρ(h) and coefficient replacements.

Abstract

The primary objective of this paper is to establish several sharp versions of improved Bohr inequality, refined Bohr-type inequality, and refined Bohr-Rogosinski inequality for the class of $K$-quasiconformal sense-preserving harmonic mappings $f=h+\overline{g}$ in the unit disk $\mathbb{D} := \{z\in\mathbb{C} : |z| < 1\}$. In order to achieve these objectives, we employ the non-negative quantity $S_\rho(h)$ and the concept of replacing the initial coefficients of the majorant series by the absolute values of the analytic function and its derivative, as well as other various settings. Moreover, we obtain the sharp Bohr-Rogosinski radius for harmonic mappings in the unit disk by replacing the bounding condition on the analytic function $h$ with the half-plane condition.