On the Theorem of Wan for K-Quasiconformal Hyperbolic Harmonic Self Mappings of the Unit Disk

Miljan Knežević

We give a new glance to the theorem of Wan (Theorem 1.1) which is related to the hyperbolic bi-Lipschicity of the $K$-quasiconformal, $K\geq 1$, hyperbolic harmonic mappings of the unit disk $\mathbb{D}$ onto itself. Especially, if $f$ is such a mapping and $f(0) = 0$, we obtained that the following double inequality is valid $2|z|/(K+1) \leq |f(z)| \leq \sqrt{K|z|}$, whenever $z\in\mathbb{D}$.