Extension of conformal mappings and hyperbolic metrics (Q913981)

In this paper the following theorems are proved: 1. If \(f\in \Sigma\), then for every t in the unit disk \[ f_ t=tf(z/t)\in \Sigma (| t|^ 2). \] If \(f\in \Sigma (k)\) \((0<k<1)\), then \(f_ t\in \Sigma (k| t|)\). Here the classes \(\Sigma\) and \(\Sigma\) (k) are defined as usual. 2. Let L be the image of the unit circle \(\{| z| =1\}\) under a conformal mapping of \(\{r<| z| <R\}\). Then L allows a K- quasiconformal mapping where \[ K\leq [1+(rR)^ 2]/(r^ 2+R^ 2). \]