A generalization of the Schwarz-Carathéodory reflection principle and spaces of pseudo-metrics (Q2710565)

Let \(\mathbf{D}\) denote the unit disk and let \(\Omega \subset \mathbf{C}\) be a simply connected domain. If \(w(z)\) is analytic in \(\Omega, w(\Omega) \subset \mathbf{D}\), then NEWLINE\[NEWLINE w^*(z)={|w'(z)|\over 1-|w(z)|^2} \qquad (z \in \Omega) NEWLINE\]NEWLINE denotes a pseudo-metric of hyperbolic type.NEWLINENEWLINENEWLINEThe classical Schwarz-Carathéodory reflection principle can be put in the form: Let \(I \subset \partial\Omega\) be an open, analytic and free boundary arc and let \(0 \neq w(z)\) be analytic in \(\Omega\). Then \(w(z)\) is analytic in \(\Omega \cup I\) if, and only if, there exists \(v(z)\), analytic in \(I\), such that NEWLINE\[NEWLINE \lim_{z \rightarrow \zeta} \left|{w(z) \over v(z)}\right|\rightarrow 1 \qquad (\zeta \in I). NEWLINE\]NEWLINE The autors prove a similar statement when \(w(z)\) is replaced by the hyperbolic \(w^*(z)\), and they give structural properties of spaces of pseudo-metrics.