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arXiv · 0711.0170

Spherical and hyperbolic lengths of images of arcs

T. K. Carne
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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 $ρ$ is then $o(ρ^1/2)$. 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.

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T. K. Carne. 2007-11-01. Spherical and hyperbolic lengths of images of arcs. https://arxiv.org/abs/0711.0170

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