Bohr-Type Inequalities for Harmonic Mappings with a Multiple Zero at the Origin

In this paper, we first determine Bohr’s inequality for the class of harmonic mappings f= h+ g¯ in the unit disk D, where either both h(z)=∑n=0∞apn+mzpn+m and g(z)=∑n=0∞bpn+mzpn+m are analytic and bounded in D, or satisfies the condition | g′(z) | ≤ d| h′(z) | in D\ { 0 } for some d∈ [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.