A class of norms on the spaces of Schwarzian derivatives and its applications (Q1318916)

Let \(\varphi\) be analytic in the plane domain \(D\). For \(1\leq A<\infty\), the author considers the norms given by \[ A^ 2\sup\sup\{|\varphi(z)|:z\in D,| z-c|<r\} \] where the first supremum is taken over all \(c\in D\) and \(r>0\) such that \(\{| z- c|<Ar\}\subset D\). These norms have a close connection to the local injectivity and quasiconformal extensibility of a function \(f\) for the case that \(\varphi\) is the Schwarzian derivative of \(f\).