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 $h(0)=h^{\prime }(0)-1=0$ and $g(0)=g^{\prime }(0)=0$ with the second complex dilatation $ω:\,\ID\rightarrow \ID$, $g^{\prime }(z)=\omega (z)h^{\prime }(z)$. 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 $\ID$.

16 pages, 12 figures; To be published in the Proceedings of the International Conference on Complex Analysis and Dynamical Systems VI