# zbMATH — the first resource for mathematics

On bi-Lipschitz type inequalities for quasiconformal harmonic mappings. (English) Zbl 1123.30010
Recently M. Pavlović [Ann. Acad. Sci. Fenn. Math. 27, No. 2, 365–372 (2002; Zbl 1017.30014)] proved that any quasiconformal and harmonic self mapping $$F$$ of the unit disk $$D$$ is bi-Lipschitz w.r.t. the Euclidean metric. One aim of this paper is to improve Pavlović’s results. Let $$f$$ be the boundary valued fuction for $$F$$ and let $$T$$ denote the unit circumference. The authors first consider operators $$V[f]$$ and $$V^*[f]$$ defined as follows:
\begin{aligned} V[f]&= \lim_{\varepsilon\to 0}(2\pi)^{-1}\int_{T \setminus T(z,\varepsilon)}|f(u)-f(z)|^2|u-z|^{-2}|du|,\\ V^*[f]&= \lim_{\varepsilon\to 0}(2\pi)^{-1}\int_{T \setminus T(z,\varepsilon)} \operatorname{Im}[f(u)\overline{f(z)}]|u-z|^{-2}|du|,\;z\in T, \end{aligned}
and show that they satisfy certain inequalities. This is then used to obtain bi-Lipschitz type estmations for $$f$$ provided $$F(0)=0$$. The results are asymptotically sharp as $$K\to 1$$ and show that for sufficiently small $$K$$, $$F$$ behaves like a rotation.

##### MSC:
 30C55 General theory of univalent and multivalent functions of one complex variable 30C62 Quasiconformal mappings in the complex plane
Zbl 1017.30014