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 \(\mathbb {D}\), 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 \(\mathbb {D}\), or satisfies the condition \(|g'(z)|\le d|h'(z)|\) in \(\mathbb {D}\backslash \{0\}\) 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.