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 $$ g&#39;(z)=ω(z)h&#39;(z)+ψ(z),$$ where $ω$ and $ψ$ are analytic functions in $\mathbb{D}$ with $\|ω\|_{\infty}\leq k<1$ and $\|ψ\|_{\infty}\leq M$. 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.