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On the quasi-isometries of harmonic quasiconformal mappings. (English) Zbl 1116.30010
The authors study mappings which are harmonic with respect to the Euclidean or Poincaré metric and, in addition, quasiconformal. In both cases they prove a variant of the Ahlfors-Schwarz lemma. For instance they prove the following result. Let $$f: U \to U$$ be a $$k$$-quasiconformal Euclidean harmonic mapping onto the unit disk $$U$$ in $$R^2.$$ Then for all $$z_1,z_2 \in U$$ $d_h(f(z_1),f(z_2))\leq \frac{1+k}{1-k} d_h(z_1,z_2)$ where $$d_h$$ is the hyperbolic metric of $$U.$$

MSC:
 30C62 Quasiconformal mappings in the complex plane 30C65 Quasiconformal mappings in $$\mathbb{R}^n$$, other generalizations
Keywords:
quasiconformal map; harmonic map
Full Text:
References:
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