Sur allures de la densité de Poincaré et ses dérivées au voisinage d'un point frontière (Q1316515)

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.