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Bohr-type inequalities for harmonic mappings with a multiple zero at the origin (2103.09403v1)
Published 17 Mar 2021 in math.CV
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.