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The Poincaré metric and isoperimetric inequalities for hyperbolic polygons

Authors: Roger W. Barnard, Petros Hadjicostas and Alexander Yu. Solynin
Journal: Trans. Amer. Math. Soc. 357 (2005), 3905-3932
MSC (2000): Primary 30C75; Secondary 33B15
Published electronically: May 20, 2005
MathSciNet review: 2159693
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Abstract: We prove several isoperimetric inequalities for the conformal radius (or equivalently for the Poincaré density) of polygons on the hyperbolic plane. Our results include, as limit cases, the isoperimetric inequality for the conformal radius of Euclidean $n$-gons conjectured by G. Pólya and G. Szegö in 1951 and a similar inequality for the hyperbolic $n$-gons of the maximal hyperbolic area conjectured by J. Hersch. Both conjectures have been proved in previous papers by the third author.

Our approach uses the method based on a special triangulation of polygons and weighted inequalities for the reduced modules of trilaterals developed by A. Yu. Solynin. We also employ the dissymmetrization transformation of V. N. Dubinin. As an important part of our proofs, we obtain monotonicity and convexity results for special combinations of the Euler gamma functions, which appear to have a significant interest in their own right.

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Additional Information

Roger W. Barnard
Affiliation: Department of Mathematics and Statistics, Texas Tech University, Box 41042, Lubbock, Texas 79409

Petros Hadjicostas
Affiliation: Department of Mathematics and Statistics, Texas Tech University, Box 41042, Lubbock, Texas 79409

Alexander Yu. Solynin
Affiliation: Steklov Institute of Mathematics at St. Petersburg, Russian Academy of Sciences, Fontanka 27, St. Petersburg, 191011, Russia
Address at time of publication: Department of Mathematics and Statistics, Texas Tech University, Box 41042, Lubbock, Texas 79409

Keywords: Isoperimetric inequality, hyperbolic geometry, Poincar\'{e} metric, polygon, conformal radius, absolutely monotonic function, Euler gamma function
Received by editor(s): March 11, 2003
Published electronically: May 20, 2005
Additional Notes: This paper was finalized during the third author’s visit to Texas Tech University, 2001–2002. This author thanks the Department of Mathematics and Statistics of this University for the wonderful atmosphere and working conditions during his stay in Lubbock. The research of the third author was supported in part by the Russian Foundation for Basic Research, grant no. 00-01-00118a.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.