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On quasicircles and asymptotically conformal curves. (English. Russian original) Zbl 0822.30020
Russ. Acad. Sci., Dokl., Math. 47, No. 3, 563-566 (1993); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 330, No. 5, 546-548 (1993).
A Jordan curve \(\Gamma \subset \mathbb{C}\) is called asymptotically conformal if \[ \lim_{| w_ 1 - w_ 2 | \to 0} {| w_ 1 - w | + | w - w_ 2 | \over | w_ 1 - w_ 2 |} = 1 \] where \(w\) is an arbitrary point of the smaller arc of \(\Gamma\) with endpoints \(w_ 1, w_ 2\). The authors give now an interesting characterization of these curves using the concept of asymptotical homogeneity. A mapping \(f : D \to \mathbb{C}\) is called asymptotically homogeneous at \(z \in D\) \((D \subset \mathbb{C}\) domain) if, for each \(\zeta \in \mathbb{C}\), \[ \lim_{\eta \to 0} {f(z + \zeta \eta) - f(z) \over f(z + \eta) - f(z)} = \zeta. \] The main result is now that the following assertions are equivalent
1. \(\Gamma\) is an asymptotically conformal curve.
2. There exists a quasiconformal mapping \(f : \mathbb{C} \to \mathbb{C}\) with \(\Gamma = f(\mathbb{T})\) where \(\mathbb{T}\) denotes the unit circle, such that \(f\) is uniformly asymptotically homogeneous on \(\mathbb{T}\).
3. Any conformal mapping of the unit disk \(\mathbb{D}\) onto the interior of \(\Gamma\) is likewise uniformly asymptotically homogeneous on \(\mathbb{D}\). The authors give some more related conditions that are also equivalent. The proofs are omitted but can be found elsewhere. [Same authors: Complex Variables, Theory Appl. 25, No. 4, 357-366 (1994; Zbl 0819.30011)].
30C62 Quasiconformal mappings in the complex plane