Computational Methods and Function Theory 2 (2002), No. 2, 337--351
Copyright Heldermann Verlag 2002
Diego Mejía
dmejia@perseus.unalmed.edu.co, Universidad Nacional, Departamento de Matemáticas, A.A. 3840 Medellin, Colombia.
Christian Pommerenke
pommeren@math.tu-berlin.de, Technische Universität Berlin, Institut für Mathematik, MA 8-2, 10623 Berlin, Germany.
Let $f$ be a conformal map of the unit disk $\mathbb{D}$ into $\hat{\mathbb{C}}$ and let $$ Q_{f}(z,\zeta) = \frac{(1-|z|^2) |f'(z)| (1-|\zeta|^2) |f'(\zeta)|} {|f(z) - f(\zeta)|^2} \lambda_{\mathbb{D}}(z,\zeta)^2, $$ where $\lambda_{\mathbb{D}}$ denotes the hyperbolic distance. We introduce the family $\ML$ of all conformal maps $f$ for which $Q_{f}(z,\zeta)$ remains bounded. It contains all maps $f$ that have a quasi-conformal extension to $\hat{\mathbb{C}}$ but also some functions for which $f(\mathbb{D})$ has outward-pointing cusps. We show that $f$ has a continuous extension to $\overline{\mathbb{D}}$ and study multiple boundary points and the Schwarzian derivative.
Keywords: Möbius-invariant, conformal map, quasi-conformal extension, multiple points, cusp, Schwarzian derivative.
MSC 2000: Primary 30C55; Secondary 30C45, 30C62.
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