Landau’s theorem for planar harmonic mappings

Abstract

Chen, Gauthier and Hengartner obtained two versions of Landau’s theorem for bounded planar harmonic mappings. Later, Dorff and Nowak improved their estimates. Furthermore, Grigoryan and Huang independently improved their results. In this note, we improve these last results by obtaining better coefficient estimates for bounded and normalized planar harmonic mappings. In particular, our theorems are sharp when M=1, which are consistent with Landau’s theorem when M=1.