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

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 $|g^{\prime }(z)|\le d|h^{\prime }(z)|$ 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.

20 pages; The article has appeared in Mediterranean Journal of Mathematics (2021)

References in corpus (1)

• Bohr operator on analytic functions