Radial projection and the Poincaré metric
Let us consider domains on the Riemann sphere \(\overline C\) which omit at least three points. The unit disk \(U=\{z:|z|<1\}\) is the universal covering surface of any such domain \(D\). If \(\pi:U\to D\) is the universal covering, then the Poincaré metric \(\lambda_D\) for \(D\) is given by \[ \lambda_D \bigl(\pi(z) \bigr)\bigl|\pi'(z)\bigr|= {1\over 1-|z |^2}, \quad z\in U,\;w=\pi(z) \neq\infty \] and if \(w=\infty\), then one sets \(\lambda_D(\infty)= \lambda_D^{-1}(0)\), where \(D^{-1}=\{z:z^{-1}\in D\}\). Let \(E\subset\overline C\) and let \(E_{\text{cire}}\) denote radial projection of \(E\) into the unit circle \(T=T_1\), where \(\text{Tr}= \{z:|z|=r\}\), \(r>0\). ``One of the main purposes of this note is to derive some new estimates for the Poincaré metric \(\lambda_D\). These estimates are given in terms of the radial projection \(E_{\text{cire}}\) of the set \(E=\overline C\setminus D\) onto the circle \(T\).
scientific article; zbMATH DE number 1364721