### Abstract

We prove several isoperimetric inequalities for the conformal radius (or equivalently for the Poincar\'{e} density)
of polygons on the hyperbolic plane. Our results include, as the limit cases, the isoperimetric inequality for the conformal radius
of Euclidean $n$-gons conjectured by G.~P\'{o}lya and G.~Szeg\"{o} 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.

Original language | English |
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Pages (from-to) | 3905-3932 |

Journal | Transactions of the American Mathematical Society |

State | Published - Oct 2005 |

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## Cite this

Barnard, R., Solynin, A., & Hadjicostas, P. (2005). The Poincare metric and isoperimetric inequalities for hyperbolic polygons.

*Transactions of the American Mathematical Society*, 3905-3932.