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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
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