Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The Poincaré metric and isoperimetric inequalities for hyperbolic polygons


Authors: Roger W. Barnard, Petros Hadjicostas and Alexander Yu. Solynin
Journal: Trans. Amer. Math. Soc. 357 (2005), 3905-3932
MSC (2000): Primary 30C75; Secondary 33B15
Published electronically: May 20, 2005
MathSciNet review: 2159693
Full-text PDF Free Access

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 $n$-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 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 [Enhancements On Off] (What's this?)

  • 1. Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. MR 0357743
  • 2. Horst Alzer, On some inequalities for the gamma and psi functions, Math. Comp. 66 (1997), no. 217, 373–389. MR 1388887, 10.1090/S0025-5718-97-00807-7
  • 3. Glen D. Anderson, Mavina K. Vamanamurthy, and Matti K. Vuorinen, Conformal invariants, inequalities, and quasiconformal maps, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1997. With 1 IBM-PC floppy disk (3.5 inch; HD); A Wiley-Interscience Publication. MR 1462077
  • 4. George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958
  • 5. Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vols. I, II, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR 0058756
  • 6. Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1995. Corrected reprint of the 1983 original. MR 1393195
  • 7. K. Bezdek, Ein elementarer Beweis für die isoperimetrische Ungleichung in der Euklidischen und hyperbolischen Ebene, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 27 (1984), 107–112 (1985) (German). MR 823098
  • 8. Roberto Bonola, Non-Euclidean geometry, a critical and historical study of its developments, Dover Publications, Inc., New York, 1955. Translation with additional appendices by H. S. Carslaw; Supplement containing the G. B. Halsted translations of “The science of absolute space” by John Bolyai and “The theory of parallels” by Nicholas Lobachevski. MR 0070197
  • 9. Yu. D. Burago and V. A. Zalgaller, Geometric inequalities, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 285, Springer-Verlag, Berlin, 1988. Translated from the Russian by A. B. Sosinskiĭ; Springer Series in Soviet Mathematics. MR 936419
  • 10. V. N. Dubinin, Change of harmonic measure in symmetrization, Mat. Sb. (N.S.) 124(166) (1984), no. 2, 272–279 (Russian). MR 746071
  • 11. V. N. Dubinin, Symmetrization in the geometric theory of functions of a complex variable, Uspekhi Mat. Nauk 49 (1994), no. 1(295), 3–76 (Russian); English transl., Russian Math. Surveys 49 (1994), no. 1, 1–79. MR 1307130, 10.1070/RM1994v049n01ABEH002002
  • 12. Peter L. Duren, Univalent functions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. MR 708494
  • 13. William Feller, An introduction to probability theory and its applications. Vol. II., Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0270403
  • 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, Vol. 26, American Mathematical Society, Providence, R.I., 1969. MR 0247039
  • 17. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London-Toronto, Ont., 1980. Corrected and enlarged edition edited by Alan Jeffrey; Incorporating the fourth edition edited by Yu. V. Geronimus [Yu. V. Geronimus] and M. Yu. Tseytlin [M. Yu. Tseĭtlin]; Translated from the Russian. MR 582453
  • 18. Marvin Jay Greenberg, Euclidean and non-Euclidean geometries, 3rd ed., W. H. Freeman and Company, New York, 1993. Development and history. MR 1261866
  • 19. G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge, at the University Press, 1952. 2d ed. MR 0046395
  • 20. E. W. Hobson, The theory of functions of a real variable and the theory of Fourier’s series. Vol. II, Dover Publications, Inc., New York, N.Y., 1958. MR 0092829
  • 21. James A. Jenkins, Univalent functions and conformal mapping, Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge, Heft 18. Reihe: Moderne Funktionentheorie, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958. MR 0096806
  • 22. Werner von Koppenfels and Friedmann Stallmann, Praxis der konformen Abbildung, Die Grundlehren der mathematischen Wissenschaften, Bd. 100, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1959 (German). MR 0107698
  • 23. Hsu-Tung Ku, Mei-Chin Ku, and Xin-Min Zhang, Isoperimetric inequalities on surfaces of constant curvature, Canad. J. Math. 49 (1997), no. 6, 1162–1187. MR 1611644, 10.4153/CJM-1997-057-x
  • 24. Reiner Kühnau, Zum konformen Radius bei nullwinkligen Kreisbogendreiecken, Mitt. Math. Sem. Giessen 211 (1992), 19–24 (German). MR 1188834
  • 25. Olli Lehto, Univalent functions and Teichmüller spaces, Graduate Texts in Mathematics, vol. 109, Springer-Verlag, New York, 1987. MR 867407
  • 26. Albert W. Marshall and Ingram Olkin, Inequalities: theory of majorization and its applications, Mathematics in Science and Engineering, vol. 143, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR 552278
  • 27. John Milnor, Hyperbolic geometry: the first 150 years, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 1, 9–24. MR 634431, 10.1090/S0273-0979-1982-14958-8
  • 28. G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, no. 27, Princeton University Press, Princeton, N. J., 1951. MR 0043486
  • 29. Christian Pommerenke, Univalent functions, Vandenhoeck & Ruprecht, Göttingen, 1975. With a chapter on quadratic differentials by Gerd Jensen; Studia Mathematica/Mathematische Lehrbücher, Band XXV. MR 0507768
  • 30. B. A. Rosenfeld, A history of non-Euclidean geometry, Studies in the History of Mathematics and Physical Sciences, vol. 12, Springer-Verlag, New York, 1988. Evolution of the concept of a geometric space; Translated from the Russian by Abe Shenitzer. MR 959136
  • 31. A. Yu. Solynin, Solution of the Pólya-Szegő isoperimetric problem, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 168 (1988), no. Anal. Teor. Chisel i Teor. Funktsii. 9, 140–153, 190 (Russian); English transl., J. Soviet Math. 53 (1991), no. 3, 311–320. MR 982489, 10.1007/BF01303655
  • 32. A. Yu. Solynin, Isoperimetric inequalities for polygons and dissymetrization, Algebra i Analiz 4 (1992), no. 2, 210–234 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 4 (1993), no. 2, 377–396. MR 1182401
  • 33. A. Yu. Solynin, Some extremal problems for circular polygons, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 206 (1993), no. Issled. po Linein. Oper. i Teor. Funktsii. 21, 127–136, 176 (Russian, with English and Russian summaries); English transl., J. Math. Sci. 80 (1996), no. 4, 1956–1961. MR 1255321, 10.1007/BF02367011
  • 34. A. Yu. Solynin, Some extremal problems on the hyperbolic polygons, Complex Variables Theory Appl. 36 (1998), no. 3, 207–231. MR 1671474
  • 35. A. Yu. Solynin, Moduli and extremal metric problems, Algebra i Analiz 11 (1999), no. 1, 3–86 (Russian); English transl., St. Petersburg Math. J. 11 (2000), no. 1, 1–65. MR 1691080
  • 36. Alexander Yu. Solynin and Victor A. Zalgaller, An isoperimetric inequality for logarithmic capacity of polygons, Ann. of Math. (2) 159 (2004), no. 1, 277–303. MR 2052355, 10.4007/annals.2004.159.277
  • 37. Saul Stahl, The Poincaré half-plane, Jones and Bartlett Publishers, Boston, MA, 1993. A gateway to modern geometry. MR 1217085

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: http://dx.doi.org/10.1090/S0002-9947-05-03946-2
Keywords: Isoperimetric inequality, hyperbolic geometry, Poincar\'{e} metric, polygon, conformal radius, absolutely monotonic function, Euler gamma function
Received by editor(s): March 11, 2003
Published electronically: 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.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.