TY - JOUR

T1 - The poincaré metric and isoperimetric inequalities for hyperbolic polygons

AU - Barnard, Roger W.

AU - Hadjicostas, Petros

AU - Solynin, Alexander Yu

PY - 2005/10

Y1 - 2005/10

N2 - 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.

AB - 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.

KW - Absolutely monotonic function

KW - Conformal radius

KW - Euler gamma function

KW - Hyperbolic geometry

KW - Isoperimetric inequality

KW - Poincaré metric

KW - Polygon

UR - http://www.scopus.com/inward/record.url?scp=26444466691&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-05-03946-2

DO - 10.1090/S0002-9947-05-03946-2

M3 - Article

AN - SCOPUS:26444466691

VL - 357

SP - 3905

EP - 3932

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 10

ER -