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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

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The Poincaré metric and isoperimetric inequalities for hyperbolic polygons
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by Roger W. Barnard, Petros Hadjicostas and Alexander Yu. Solynin PDF
Trans. Amer. Math. Soc. 357 (2005), 3905-3932 Request permission

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
  • MR Author ID: 31355
  • Email: barnard@math.ttu.edu
  • Petros Hadjicostas
  • Affiliation: Department of Mathematics and Statistics, Texas Tech University, Box 41042, Lubbock, Texas 79409
  • Email: phadjico@math.ttu.edu
  • 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
  • MR Author ID: 206458
  • Email: solynin@math.ttu.edu
  • 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.
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 357 (2005), 3905-3932
  • MSC (2000): Primary 30C75; Secondary 33B15
  • DOI: https://doi.org/10.1090/S0002-9947-05-03946-2
  • MathSciNet review: 2159693