Bohr-type inequalities for harmonic mappings with a multiple zero at the origin

TL;DR

This paper establishes Bohr's inequalities for harmonic mappings with multiple zeros at the origin, including classes with bounded components and specific derivative conditions, with many results proven to be sharp.

Contribution

It extends Bohr's inequality to harmonic mappings with multiple zeros and specific structural conditions, providing sharp bounds and new classes of functions.

Findings

Derived Bohr's inequalities for harmonic mappings with multiple zeros

Established sharp bounds for these inequalities

Extended results to p-symmetric harmonic mappings

Abstract

In this paper, we first determine Bohr's inequality for the class of harmonic mappings $f=h+\overline{g}$ in the unit disk $\ID$, where either both $h(z)=\sum_{n=0}^{\infty}a_{pn+m}z^{pn+m}$ and $g(z)=\sum_{n=0}^{\infty}b_{pn+m}z^{pn+m}$ are analytic and bounded in $\ID$, or satisfies the condition $|g'(z)|\leq d|h'(z)|$ in $\ID\backslash \{0\}$ for some $d\in [0,1]$ and $h$ is bounded. In particular, we obtain Bohr's inequality for the class of harmonic $p$-symmetric mappings. Also, we investigate the Bohr-type inequalities of harmonic mappings with a multiple zero at the origin and that most of results are proved to be sharp.