Bohr radius for locally univalent harmonic mappings

TL;DR

This paper investigates the Bohr radius for various classes of harmonic and analytic functions in the unit disk, providing precise bounds and exploring the effects of boundedness and dilatation constraints.

Contribution

It determines the Bohr radius for multiple classes of harmonic mappings under different boundedness and dilatation conditions, extending classical results to harmonic functions.

Findings

Calculated Bohr radius for harmonic mappings with bounded analytic parts.

Established Bohr radius bounds when the real part of the analytic part is bounded above.

Extended Bohr radius concepts to harmonic Bloch functions.

Abstract

We consider the class of all sense-preserving harmonic mappings $f= h+\overline{g}$ of the unit disk $\ID$, where $h$ and $g$ are analytic with $g(0)=0$, and determine the Bohr radius if any one of the following conditions holds: \bee $h$ is bounded in $\ID$. $h$ satisfies the condition ${\rm Re}\, h(z)\leq 1$ in $\mathbb{D}$ with $h(0)>0$. both $h$ and $g$ are bounded in $\ID$. $h$ is bounded and $g'(0)=0$. \eee We also consider the problem of determining the Bohr radius when the supremum of the modulus of the dilatation of $f$ in $\ID$ is strictly less than $1$. In addition, we determine the Bohr radius for the space $\mathcal B$ of analytic Bloch functions and the space ${\mathcal B}_H$ of harmonic Bloch functions. The paper concludes with two conjectures.