The Poincare metric and isoperimetric inequalities for hyperbolic polygons

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.