Bloch and Landau Type Theorems for Harmonic Mappings with Inhomogeneous Analytic Dilatation

Abstract

We study Bloch and Landau type theorems for a class of sense-preserving harmonic mappings $f=h+\overline{g}$ in the unit disk $\mathbb{D}$ satisfying the inhomogeneous analytic dilatation equation $$ g'(z)=\omega(z)h'(z)+\psi(z),$$ where $\omega$ and $\psi$ are analytic functions in $\mathbb{D}$ with $\|\omega\|_{\infty}\leq k<1$ and $\|\psi\|_{\infty}\leq M$. Here $h$ and $g$ are called analytic and co-analytic part of $f$, respectively. We first establish a Bloch type theorem for certain normalized class of harmonic functions under the condition $k+M<1$. Finally, we obtain two versions of the Landau theorem under additional assumptions: one for bounded harmonic mappings and another under the assumption that the analytic part of a harmonic functions has bounded Bloch seminorm.