John domains and a univalence criterion of Ahlfors
For a function \(f\) analytic in the unit disk \(D\), let \(Sf\) denote the Schwarzian derivative of \(f\), \(Sf=({f''\over f'})-{1\over 2}({f'' \over f'})^2\). A bounded, simply connected domain \(\Omega\) is called a John domain if there is a positive consant \(a\) such that for every crosscut \(C\) of \(\Omega\) there exists a component \(H\) of \(\Omega\setminus C\) such that \(\text{diam} H\leq a \text{diam} C\). The author proves two results. Theorem 1. If \(1<t\leq 2\) and \(f\) satisfies \[ \left| Sf-{2t(1-t) \overline z^2\over \bigl( 1-| z|^2\bigr)^2} \right|\leq {2t\over\bigl(1-| z|^2 \bigr)^2}, \] and if \(f(D)\) is bounded, then \(f(D)\) is a John domain. Theorem 2. If \(0\leq t<1\) and \(f\) satisfies \[ \left| Sf-{2t(1-t)\overline z^2\over \bigl(1-| z|^2 \bigr)^2}\right |\leq {2t\over \bigl(1-| z|^2 \bigr)^2}+ {2\pi\over \bigl(1-| z|^2\bigr)^{2t}} \left({\Gamma( {\textstyle {3\over 2}}- t)\over\Gamma (1-t)}\right)^2, \] and \(f(D)\) is a Jordan domain, then \(f(D)\) is a quasi-disk. Further, in the case \(0\leq t<{1\over 2}\), \(f(D)\) is a Jordan domain and hence a quasi-disk.
scientific article; zbMATH DE number 1185372