Extreme points method and univalent harmonic mappings

Abstract

We consider the class of all sense-preserving complex-valued harmonic mappings $f=h+\bar {g}$ defined on the unit disk $\ID$ with the normalization $h(0)=h'(0)-1=0$ and $g(0)=g'(0)=0$ with the second complex dilatation $\omega:\,\ID\rightarrow \ID$, $g'(z)=\omega (z)h'(z)$. In this paper, the authors determine sufficient conditions on $h$ and $\omega$ that would imply the univalence of harmonic mappings $f=h+\bar {g}$ on $\ID$.