Bohr's Phenomenon for Some Univalent Harmonic Functions

TL;DR

This paper extends Bohr's phenomenon to certain univalent harmonic functions, determining their Bohr radii and exploring cases with different dilatations and convexity in one direction.

Contribution

It introduces new Bohr radius results for univalent harmonic functions with various dilatations and convexity properties.

Findings

Determined Bohr radii for specific univalent harmonic functions.

Computed Bohr radius for functions convex in one direction.

Extended Bohr's phenomenon to harmonic mappings with different dilatations.

Abstract

In 1914 Bohr proved that there is an $r_0 \in(0,1)$ such that if a power series $\sum_{m=0}^\infty c_m z^m$ is convergent in the open unit disc and $|\sum_{m=0}^\infty c_m z^m|<1$ then, $\sum_{m=0}^\infty |c_m z^m|<1$ for $|z|<r_0$. The largest value of such $r_0$ is called the Bohr radius. In this article, we find Bohr radius for some univalent harmonic mappings having different dilatations and in addition, also compute Bohr radius for the functions convex in one direction.