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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)].
##### MSC:
 30C62 Quasiconformal mappings in the complex plane