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A quasiconformal extension of conformal mappings. I. (English) Zbl 0885.30016
Let $$f(z)=z+a_2z^2+\cdots$$ be regular in the unit disk $$U(|z|<1)$$ and $$f'(z)\not=0$$ in $$U$$. By constructing a Löwner chain, the authors prove that if there exists $$q\in(0,1)$$ such that $(1-|z|^2)^2|S_f(z)|/2\leq q,\;z\in U,$ then $$f$$ is univalent in $$U$$ and has a $$k$$-quasiconformal extension to $$U^-(|z|>1)$$, where $$S_f(z)$$ is the Schwarzian derivative of $$f$$ and $$k=(1+q)/(1-q)$$.
Reviewer: L.Liu (Harbin)
##### MSC:
 30C62 Quasiconformal mappings in the complex plane