Sur allures de la densité de Poincaré et ses dérivées au voisinage d'un point frontière
Suppose \(\Omega\) is a hyperbolic region in the complex plane \(\mathbb{C}\) and \(a\in\mathbb{C}\) is an isolated boundary point of \(\Omega\). The author investigates the behavior of the density \(\lambda_ \Omega(z)\) of the hyperbolic, or Poincaré, metric and certain of its partial derivatives in a punctured neighborhood of \(a\). For instance, if \(\Phi_ n(z)=| z-a|^ n\log| z-a|^{-1}\), then \(\Phi_ 1(z)\lambda_ \Omega(z)\to 1/2\) when \(z\to a\). Also, the author shows that the quantities \(\Phi_ 2(z)|{\partial\lambda_ \Omega(z)\over\partial z}|\), \(\Phi_ 3(z)|{\partial^ 2\lambda_ \Omega(z)\over\partial z^ 2}|\) and \(\Phi_ 3(z)|{\partial^ 2\lambda_ \Omega(z)\over\partial\overline z\partial z}|\) are all bounded as \(z\to a\). These orders of growth at an isolated boundary point are exact for the density of the hyperbolic metric on the punctured disk \(\{z:0<| z|<1\}\). In addition, the author considers the behavior of \(\lambda_ \Omega(z)\) near an arbitrary boundary point. Finally, the author obtains the exact growth rate for the radius of univalence of the hyperbolic metric near an isolated boundary point.
scientific article; zbMATH DE number 515595