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Proceedings of the London Mathematical Society Advance Access originally published online on March 17, 2008
Proceedings of the London Mathematical Society 2008 97(1):183-208; doi:10.1112/plms/pdm060
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© 2008 London Mathematical Society

Euclidean and hyperbolic lengths of images of arcs

A. F. Beardon and T. K. Carne

University of Cambridge
Department of Pure Mathematics and
Mathematical Statistics
Wilberforce Road
Cambridge CB3 0WB
United Kingdom
afb@dpmms.cam.ac.uk

Received 24 July 2003. Revision received 24 August 2007.

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.

2000 Mathematics Subject Classification 30F45, 30C45, 30C25.


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