Ahlfors-Weill extensions of conformal mappings and critical points of the Poincaré metric
Let \(f\) be an analytic function in the unit disk \(\mathbb{D}\) such that the Schwarzian derivative \(S_f\) satisfies \[ \bigl( 1 - |z |^2 \bigr)^2 \bigl |S_f(z) \bigr |\leq 2. \tag{*} \] Nehari has shown that \(f\) is univalent in \(\mathbb{D}\) while, by Ahlfors and Weill, the function \(f\) has a quasiconformal extension to \(\mathbb{C}\) if the right-hand side of \((*)\) is replaced by any number \(< 2\). The extension to \(\{|z |> 1\}\) is given by \[ f \left( {1 \over \overline z} \right) + \bigl( |z |^2 - 1 \bigr) f' \left( {1 \over z} \right) \left / \left[ z - {1 \over 2} \bigl( |z |^2 - 1 \bigr) f'' \left( {1 \over z} \right) \right. \left / f' \left( {1 \over 2} \right) \right. \right]. \tag{**} \] The main result of the authors is that, rather surprisingly, the formula \((**)\) always gives a homeomorphic extension whenever the (weaker) condition \((*)\) is satisfied. Another useful result is that \[ \bigl |f'' (z)/f'(z) \bigr |\leq 2 |z |/ \bigl( 1 - |z |^2 \bigr) \] if \(f\) satisfies \((*)\) and is normalized by \(f(0) = f''(0) = 0\), \(f'(0) = 1\).
scientific article; zbMATH DE number 745328