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The Quarterly Journal of Mathematics Advance Access published online on March 3, 2009
The Quarterly Journal of Mathematics, doi:10.1093/qmath/hap007
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© The author 2009. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

STRETCHING CONVEX DOMAINS AND THE HYPERBOLIC METRIC

Rodrigo Bañuelos

Department of Mathematics, Purdue University, Indiana, USA

Tom Carroll{dagger}

Department of Mathematics, National University of Ireland, Cork, Ireland

{dagger} Corresponding author. E-mail: t.carroll{at}ucc.ie

Received 16 August 2008; revised 19 January 2009
  Abstract

It is shown that the log-convexity of the density of the hyperbolic metric in a convex planar domain leads to a pointwise comparison between the density of the hyperbolic metric in a convex domain D and that in a domain obtained by stretching D. Applications of this result are given, including estimates for the density of the hyperbolic metric in the domain interior to an ellipse and a lower bound for the density of the hyperbolic metric in a convex domain in terms of the density in a comparison strip. Connections are made with the convexity of related functions on convex regions in space.


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