TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 357, Number 10, Pages 3905–3932 S 0002-9947(05)03946-2 Article electronically published on May 20, 2005 THE POINCAR ´ E METRIC AND ISOPERIMETRIC INEQUALITIES FOR HYPERBOLIC POLYGONS ROGER W. BARNARD, PETROS HADJICOSTAS, AND ALEXANDER YU. SOLYNIN Abstract. We prove several isoperimetric inequalities for the conformal ra- dius (or equivalently for the Poincar´ e 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. P´olya and G. Szeg¨o in 1951 and a similar inequality for the hyperbolic n-gons of the maximal hy- perbolic 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 poly- gons 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. 1. Introduction and main results Hyperbolic geometry, an important and rich field of modern mathematics, has almost a two hundred year history; see [8, 18, 27, 30]. Its standard planar model suggested by H. Poincar´ e in 1882 – see [27, 37] for interesting historical discussions – can be realized as the unit disk U = {z : |z| < 1} in the complex plane C supplied with the Poincar´ e metric (1.1) dσ U (z)= |dz| 1 −|z| 2 , z ∈ U. Some authors [6, 37] define dσ U with an extra factor 2 in the right-hand side of (1.1) enjoying the advantage of having a metric of constant curvature −1 instead of −4 in our case. However, we prefer the form (1.1), since this leads to Euclidean geometry with a standard unit of length as z approaches the origin, which simplifies many of our formulas. The hyperbolic distance ρ(z 1 ,z 2 ) between any two points Received by the editors March 11, 2003. 2000 Mathematics Subject Classification. Primary 30C75; Secondary 33B15. Key words and phrases. Isoperimetric inequality, hyperbolic geometry, Poincar´ e metric, poly- gon, conformal radius, absolutely monotonic function, Euler gamma function. 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. c 2005 American Mathematical Society Reverts to public domain 28 years from publication 3905 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use