TY - JOUR
T1 - The poincaré metric and isoperimetric inequalities for hyperbolic polygons
AU - Barnard, Roger W.
AU - Hadjicostas, Petros
AU - Solynin, Alexander Yu
N1 - Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.
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 -