The Distortion Theorems for Harmonic Mappings with Negative Coefficient Analytic Parts

Abstract

Some sharp estimates for coefficients, distortion and the growth order are obtained for harmonic mappings $f ∈ TL^α_H$ which are locally univalent harmonic mappings in the unit disk $\mathbb{D}:=\{z:|z| < 1\}$. Moreover, denoting the subclass $TS^α_H$ of the normalized univalent harmonic mappings, we also estimate the growth of $|f|,$ $f ∈ TS^α_H,$ and their covering theorems.