Euclidean and hyperbolic lenghs of images of arcs

TL;DR

This paper investigates the Euclidean length of images of radial segments under analytic functions in the unit disc, using hyperbolic geometry to derive new estimates and proofs, and explores related new questions.

Contribution

It introduces a hyperbolic geometry-based approach to estimate Euclidean lengths of images of radial segments, providing new proofs and extending existing results.

Findings

New estimates for Euclidean length of images under analytic functions

Alternative proofs of known estimates using hyperbolic geometry

Addresses new questions arising from hyperbolic geometric methods

Abstract

Let $f$ be a function that is analytic in the unit disc. We give new estimates, and new proofs of existing estimates, of the Euclidean length of the image under $f$ of a radial segment in the unit disc. Our methods are based on the hyperbolic geometry of plane domains, and we address some new questions that follow naturally from this approach.