|
The Poincaré metric and isoperimetric inequalities for hyperbolic polygons
Author(s):
Roger
W.
Barnard;
Petros
Hadjicostas;
Alexander
Yu.
Solynin
Journal:
Trans. Amer. Math. Soc.
357
(2005),
3905-3932.
MSC (2000):
Primary 30C75;
Secondary 33B15
Posted:
May 20, 2005
Retrieve article in:
PDF DVI PostScript
Abstract |
References |
Similar articles |
Additional information
Abstract:
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  -gons conjectured by G. Pólya and G. Szegö in 1951 and a similar inequality for the hyperbolic  -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.
References:
-
- 1.
- L. V. Ahlfors, Conformal invariants: topics in geometric function theory. McGraw-Hill, New York, 1973. MR 0357743 (50:10211)
- 2.
- H. Alzer, On some inequalities for the gamma and psi functions. Math. Comp. 66 (1997), 373-389. MR 1388887 (97e:33004)
- 3.
- G. D. Anderson, M. K. Vamanamurthy, and M. K. Vuorinen, Conformal invariants, inequalities, and quasiconformal maps. John Wiley & Sons, Inc., New York, 1997. MR 1462077 (98h:30033)
- 4.
- G. E. Andrews, R. Askey, and R. Roy, Special functions. Cambridge University Press, Cambridge, 1999. MR 1688958 (2000g:33001)
- 5.
- H. Bateman and A. Erdelyi, Higher transcendental functions. Vol. 1. McGraw-Hill, New York, 1953. MR 0058756 (15:419i)
- 6.
- A. F. Beardon, The geometry of discrete groups. Corrected reprint of the 1983 original. Graduate Texts in Mathematics, 91, Springer-Verlag, New York, 1995. MR 1393195 (97d:22011)
- 7.
- K. Bezdek, Ein elementarer Beweis für die isoperimetrische Ungleichung in der euklidischen und hyperbolischen Ebene. Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 27 (1984), 107-112. MR 0823098 (87c:52016)
- 8.
- R. Bonola, Non-Euclidean geometry, a critical and historical study of its developments. Dover Publications Inc., New York, 1955. MR 0070197 (16:1145b)
- 9.
- Yu. D. Burago and V. A. Zalgaller, Geometric inequalities. Grundlehren der Mathematischen Wissenschaften 285, Springer-Verlag, New York, 1988. MR 0936419 (89b:52020)
- 10.
- V. N. Dubinin, On the change in harmonic measure under symmetrization. Mat. Sb. 124 (166) (1984), 277-279; English transl. Math. USSR Sb. 52 (1985), 267-273. MR 0746071 (85j:30049)
- 11.
- -, Symmetrization in geometric theory of functions of a complex variable. Uspekhi Mat. Nauk 49 (1994), 3-76; English transl. Russian Math. Surveys 49(1) (1994), 1-79. MR 1307130 (96b:30054)
- 12.
- P. L. Duren, Univalent functions. Springer-Verlag, New York, 1983. MR 0708494 (85j:30034)
- 13.
- W. Feller, An introduction to probability theory and its applications, Vol. 2, 2nd ed. John Willey & Sons, Inc., 1971. MR 0270403 (42:5292)
- 14.
- L. R. Ford, Automorphic functions, 2nd ed. Chelsea, New York, 1951.
- 15.
- A. E. Fryntov, A note on a harmonic measure estimate and a conjecture of J. Velling, preprint.
- 16.
- G. M. Goluzin, Geometric theory of functions of a complex variable. Translations of Mathematical Monographs 26 AMS, Providence, R.I. (1969). MR 0247039 (40:308)
- 17.
- I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Corrected and enlarged ed., Academic Press, New York, 1980. MR 0582453 (81g:33001)
- 18.
- M. J. Greenberg, Euclidean and non-Euclidean geometries: Development and history, 3rd ed., W. H. Freeman and Co., New York, 1993. MR 1261866 (94k:51001)
- 19.
- G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, 2nd ed. Cambridge Univ. Press, Cambridge, 1952. MR 0046395 (13:727e)
- 20.
- E. W. Hobson, The theory of functions of a real variable and the theory of Fourier series, Vol. 2, 3rd ed. Dover Publications, 1927. MR 0092829 (19:1166b)
- 21.
- J. A. Jenkins, Univalent functions and conformal mapping, 2nd ed. Springer, Berlin, 1965. MR 0096806 (20:3288)
- 22.
- W. Koppenfels and F. Stallmann, Praxis der konformen Abbildung. Springer-Verlag, Berlin, 1959. MR 0107698 (21:6421)
- 23.
- H.-T. Ku, M.-C. Ku, and X.-M. Zhang, Isoperimetric inequalities on surfaces of constant curvature. Canad. J. Math. 49 (1997), 1162-1187. MR 1611644 (99i:51020)
- 24.
- R. Kühnau, Zum konformen Radius bei nullwinkligen Kreisbogendreiecken. Mitt. Math. Sem. Giessen 211 (1992), 19-24. MR 1188834 (93k:30011)
- 25.
- O. Lehto, Univalent functions and Teichmüller spaces. Springer-Verlag, New York, 1987. MR 0867407 (88f:30073)
- 26.
- A.W. Marshall and I. Olkin, Inequalities: theory of majorization and its applications. Mathematics in Science and Engineering, 143, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR 0552278 (81b:00002)
- 27.
- J. Milnor, Hyperbolic geometry: The first 150 years. Bulletin (New Series) of AMS 6 (1982), 9-24. MR 0634431 (82m:57005)
- 28.
- G. Pólya and G. Szegö, Isoperimetric inequalities in mathematical physics. Princeton University Press, Princeton, 1951. MR 0043486 (13:270d)
- 29.
- Chr. Pommerenke, Univalent functions. Vandenhoeck and Ruprecht, Göttingen, 1975. MR 0507768 (58:22526)
- 30.
- B. A. Rosenfeld, A history of non-Euclidean geometry. Springer-Verlag, New York, 1988. MR 0959136 (89k:01001)
- 31.
- A. Yu. Solynin, Solution of a Pólya-Szegö isoperimetric problem. Zap. Nauchn. Semin. LOMI 168 (1988), 140-153; English transl. J. Soviet Math. 53 (1991), 311-320. MR 0982489 (90h:30059)
- 32.
- -, Isoperimetric inequalities for polygons and dissymmetrization. Algebra i Analiz 4 (1992), no. 2, 210-234; English transl. St. Petersburg Math. J. 4 (1993), no. 2, 377-396. MR 1182401 (93i:52014)
- 33.
- -, Some extremal problems for circular polygons. Zap. Nauchn. Semin. POMI, 206 (1993), 127-136; English transl. J. Math. Sci. 80 (1996), 1956-1961. MR 1255321 (94m:30048)
- 34.
- -, Some extremal problems on the hyperbolic polygons. Complex Variables Theory Appl. 36 (1998), 207-231. MR 1671474 (99j:30030)
- 35.
- -, Modules and extremal metric problems. Algebra i Analiz 11 (1999), no. 1, 3-86; English transl. St. Petersburg Math. J. 11 (2000), no. 1, 1-65. MR 1691080 (2001b:30058)
- 36.
- A. Yu. Solynin and V. A. Zalgaller, An isoperimetric inequality for logarithmic capacity of polygons. Ann. of Math. 159 (2004), 277-303. MR 2052355 (2005a:31002)
- 37.
- S. Stahl, The Poincaré half-plane: A gateway to modern geometry. Jones and Bartlett Publishers, Boston, 1993. MR 1217085 (94c:51041)
Similar Articles:
Retrieve articles in Transactions of the American Mathematical Society
with MSC
(2000):
30C75,
33B15
Retrieve articles in all Journals with MSC
(2000):
30C75,
33B15
Additional Information:
Roger
W.
Barnard
Affiliation:
Department of Mathematics and Statistics, Texas Tech University, Box 41042, Lubbock, Texas 79409
Email:
barnard@math.ttu.edu
Petros
Hadjicostas
Affiliation:
Department of Mathematics and Statistics, Texas Tech University, Box 41042, Lubbock, Texas 79409
Email:
phadjico@math.ttu.edu
Alexander
Yu.
Solynin
Affiliation:
Steklov Institute of Mathematics at St. Petersburg, Russian Academy of Sciences, Fontanka 27, St. Petersburg, 191011, Russia
Address at time of publication:
Department of Mathematics and Statistics, Texas Tech University, Box 41042, Lubbock, Texas 79409
Email:
solynin@math.ttu.edu
DOI:
10.1090/S0002-9947-05-03946-2
PII:
S 0002-9947(05)03946-2
Keywords:
Isoperimetric inequality,
hyperbolic geometry,
Poincar\'{e} metric,
polygon,
conformal radius,
absolutely monotonic function,
Euler gamma function
Received by editor(s):
March 11, 2003
Posted:
May 20, 2005
Additional Notes:
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.
Copyright of article:
Copyright
2005,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
Comments: webmaster@ams.org