Spherical and hyperbolic lengths of images of arcs

TL;DR

This paper investigates how the Euclidean length of images of arcs behaves under analytic functions in the Dirichlet class when mapped into the hyperbolic disk or the Riemann sphere, extending known Euclidean results.

Contribution

It extends classical Euclidean length results for Dirichlet class functions to hyperbolic and spherical metrics, providing new bounds and insights.

Findings

Euclidean length of radial arcs is o(ρ^{1/2}) in the Dirichlet class

Similar length bounds hold when mapping into hyperbolic and spherical metrics

Results differ slightly depending on the target metric and domain geometry

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 $\rho$ is then $o(\rho^1/2)$. 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.