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AN UPPER BOUND FOR THE HYPERBOLIC METRIC OF A CONVEX DOMAIN

Published online by Cambridge University Press:  01 September 1997

A. D. RHODES
A. D. RHODES
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, Cambridge CB2 1SB
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Abstract

In [1], Beardon introduced the Apollonian metric α defined for any domain D in ℝ by

formula here

This metric is Möbius invariant, and for simply connected plane domains it satisfies the inequality αD[les ]2ρD, where ρD denotes the hyperbolic distance in D, and so gives a lower bound on the hyperbolic distance. Furthermore, it is shown in [1, Theorem 6.1] that for convex plane domains, the Apollonian metric satisfies ρD[les ]4 sinh [½αD], and, by considering the example of the infinite strip {x+iy[ratio ][mid ]y[mid ]<1}, that the best possible constant in this inequality is at least π. In this paper we make the following improvements.

Type
Research Article
Information
Bulletin of the London Mathematical Society , Volume 29 , Issue 5 , September 1997 , pp. 592 - 594
Copyright
© The London Mathematical Society 1997

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