Spherical and hyperbolic lengths of images of arcs

arXiv0711.0170MaRDI QIDQ6207353

T. K. Carne

Publication date: 1 November 2007

Abstract: Let $f$ be an analytic function on the unit disc which is in the Dirichlet class, so the Euclidean area of the image, counting multiplicity, is finite. The Euclidean length of a radial arc of hyperbolic length $ho$ is then $o(h{o}^1/2)$. In this note we consider the corresponding results when $f$ maps into the unit disc with the hyperbolic metric or the Riemann sphere with the spherical metric. Similar but not identical results hold.

Mathematics Subject Classification ID

Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) (30C45) Conformal metrics (hyperbolic, Poincaré, distance functions) (30F45) Covering theorems in conformal mapping theory (30C25)

This page was built for publication: Spherical and hyperbolic lengths of images of arcs

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6207353)