Extreme points method and univalent harmonic mappings

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 $ω:\,\ID\rightarrow \ID$, $g'(z)=ω(z)h'(z)$. In this paper, the authors determine sufficient conditions on $h$ and $ω$ that would imply the univalence of harmonic mappings $f=h+\bar {g}$ on $\ID$.

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