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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.

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