An Upper Bound for the Hyperbolic Metric of a Convex Domain |
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Authors: | Rhodes A D |
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Institution: | 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 defined forany domain D in Rn by
This metric is Möbius invariant, and for simply connectedplane domains it satisfies the inequality D 2 D, where D denotesthe hyperbolic distance in D, and so gives a lower bound onthe hyperbolic distance. Furthermore, it is shown in 1, Theorem6.1] that for convex plane domains, the Apollonian metric satisfies , and, by considering the example of the infinite strip {x + iy:|y|<1}, that the best possibleconstant in this inequality is at least . In this paper we makethe following improvements. |
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