The poincaré metric and isoperimetric inequalities for hyperbolic polygons

Roger W. Barnard, Petros Hadjicostas, Alexander Yu Solynin

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5 Scopus citations


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 trilateral 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.

Original languageEnglish
Pages (from-to)3905-3932
Number of pages28
JournalTransactions of the American Mathematical Society
Issue number10
StatePublished - Oct 2005


  • Absolutely monotonic function
  • Conformal radius
  • Euler gamma function
  • Hyperbolic geometry
  • Isoperimetric inequality
  • Poincaré metric
  • Polygon


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