Improved Bohr inequality for harmonic mappings

TL;DR

This paper refines the classical Bohr inequality for harmonic mappings in the unit disk, establishing improved bounds and radii under specific conditions, advancing the understanding of harmonic function behavior.

Contribution

It introduces refined Bohr inequalities for harmonic mappings with particular constraints, extending classical results with new bounds and extremal problem insights.

Findings

Established an improved Bohr inequality with a refined radius.

Derived results related to extremal problems for harmonic mappings.

Provided conditions under which the refined inequalities hold.

Abstract

Based on improving the classical Bohr inequality, we get in this paper some refined versions for a quasi-subordination family of functions, one of which is key to build our results. By means of these investigations, for a family of harmonic mappings defined in the unit disk $\D$, we establish an improved Bohr inequality with refined Bohr radius under particular conditions. Along the line of extremal problems concerning the refined Bohr radius, we derive a series of results. % in a logical way. Here the family of harmonic mappings have the form $f=h+\overline{g}$, where $g(0)=0$, the analytic part $h$ is bounded by 1 and that $|g'(z)|\leq k|h'(z)|$ in $\D$ and for some $k\in[0,1]$.