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Bieberbach’s conjecture, the de Branges and Weinstein functions and the Askey-Gasper inequality

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Abstract

The Bieberbach conjecture about the coefficients of univalent functions of the unit disk was formulated by Ludwig Bieberbach in 1916 [4]. The conjecture states that the coefficients of univalent functions are majorized by those of the Koebe function which maps the unit disk onto a radially slit plane.

The Bieberbach conjecture was quite a difficult problem, and it was surprisingly proved by Louis de Branges in 1984 [5] when some experts were rather trying to disprove it. It turned out that an inequality of Askey and Gasper [2] about certain hypergeometric functions played a crucial role in de Branges’ proof.

In this article I describe the historical development of the conjecture and the main ideas that led to the proof. The proof of Lenard Weinstein (1991) [72] follows, and it is shown how the two proofs are interrelated.

Both proofs depend on polynomial systems that are directly related with the Koebe function. At this point algorithms of computer algebra come into the play, and computer demonstrations are given that show how important parts of the proofs can be automated.

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References

  1. Aharonov, D.: On Bieberbach Eilenberg functions. Bull. Amer. Math. Soc. 76, 101–104 (1970)

    MathSciNet  MATH  Google Scholar 

  2. Askey, R., Gasper, G.: Positive Jacobi polynomial sums II. Amer. J. Math. 98, 709–737 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baernstein, A., Drasin, D., Duren, P., Marden, A. (eds.): The Bieberbach conjecture. Proceedings of the Symposium on the Occasion of the Proof. Mathematical surveys and monographs, vol. 21. American Mathematical Society, Providence, R. I. (1986)

  4. Bieberbach, L.: Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln. S.-B. Preuss. Akad. Wiss. 38, 940–955 (1916)

    Google Scholar 

  5. de Branges, L.: A proof of the Bieberbach conjecture. Acta Math. 154, 137–152 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brickman, L.: Extreme points of the set of univalent functions. Bull. Amer. Math. Soc. 76, 372–374 (1970)

    MathSciNet  MATH  Google Scholar 

  7. Brickman, L., MacGregor, T. H., Wilken, D. R.: Convex hulls of some classical families of univalent functions. Trans. Amer. Math. Soc. 156, 91–107 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  8. Carathéodory, C.: Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen. Math. Ann. 64, 95–115 (1907)

    Article  MathSciNet  MATH  Google Scholar 

  9. Carathéodory, C.: Über den Variabilitätsbereich der Fourier’schen Konstanten von positiven harmonischen Funktionen. Rend. Circ. Mat. Palermo 32, 193–217 (1911)

    MATH  Google Scholar 

  10. Carathéodory, C.: Untersuchungen über die konformen Abbildungen von festen und veränderlichen Gebieten. Math. Ann. 72, 107–144 (1912)

    Article  MathSciNet  Google Scholar 

  11. Charzyński, Z., Schiffer, M.: A new proof of the Bieberbach conjecture for the fourth coefficient. Arch. Rational Mech. Anal. 5, 187–193 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  12. Collins, G.E., Krandick, W.: An efficient algorithm for infallible polynomial complex roots isolation. In: Wang, Paul S. (ed.) Proceedings of ISSAC’92, pp. 189–194 (1992)

  13. Dieudonné, J.: Sur les fonctions univalentes. C. R. Acad. Sci. Paris 192, 1148–1150 (1931)

    Google Scholar 

  14. Duren. P.L.: Coefficients of univalent functions. Bull. Amer. Math. Soc. 83, 891–911 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  15. Duren, P.L.: Univalent functions. Grundlehren der mathematischen Wissenschaften, vol. 259. Springer-Verlag, New York-Berlin-Heidelberg-Tokyo (1983)

  16. Ekhad, S.B., Zeilberger, D.: A high-school algebra, “Formal calculus”, proof of the Bieberbach conjecture [after L. Weinstein]. In: Barcelo et al. (eds.) Jerusalem combinatorics ’93: an International Conference in Combinatorics, May 9–17, 1993, Jerusalem, Israel. Providence, RI: American Mathematical Society. Contemp. Math. 178, 113–115 (1994)

  17. Fekete, M., Szegö, G.: Eine Bemerkung über ungerade schlichte Funktionen. J. London Math. Soc. 8, 85–89 (1933)

    MATH  Google Scholar 

  18. FilzGerald, C.H.: Quadratic inequalities and coefficient estimates for schlicht functions. Arch. Rational Mech. Anal. 46, 356–368 (1972)

    Article  MathSciNet  Google Scholar 

  19. FitzGerald, C.H., Pommerenke, Ch.: The de Branges Theorem on univalent functions. Trans. Amer. Math. Soc. 290, 683–690 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  20. Friedland, S.: On a conjecture of Robertson. Arch. Rational Mech. Anal. 37, 255–261 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  21. Garabedian, P. R., Schiffer, M.: A proof of the Bieberbach conjecture for the fourth coefficient. J. Rational Mech. Anal. 4, 427–465 (1955)

    MathSciNet  Google Scholar 

  22. Gautschi, W.: Reminiscences of my involvement in de Branges’s proof of the Bieberbach conjecture. In: Baernstein, Drasin, Duren, Marden (Eds): The Bieberbach conjecture (West Lafayette, Ind., 1985), Math. Surveys Monogr. 21, Amer. Math. Soc., Providence, RI, pp. 205–211 (1986)

  23. Gong, S.: The Bieberbach conjecture. Studies in Advanced Mathematics, vol. 12. American Mathematical Society, Providence, R.I. (1999)

  24. Gronwall, T.H.: Some remarks on conformal representation. Ann. of Math. 16, 72–76 (1914–1915)

    Article  MathSciNet  Google Scholar 

  25. Grunsky, H.: Koeffizientenbedingungen für schlicht abbildende meromorphe Funktionen. Math. Z. 45, 29–61 (1939)

    Article  MathSciNet  Google Scholar 

  26. Hamilton, D.H.: Extremal boundary problems. Proc. London Math. Soc. (3) 56, 101–113 (1988)

    MathSciNet  MATH  Google Scholar 

  27. Hayman, W.K.: The asymptotic behaviour of p-valent functions. Proc. London Math. Soc. (3) 5, 257–284 (1955)

    MathSciNet  MATH  Google Scholar 

  28. Hayman, W.K., Hummel, J.A.: Coefficients of powers of univalent functions. Complex Variables 7, 51–70 (1986)

    MathSciNet  MATH  Google Scholar 

  29. Heine, E.: Handbuch der Kugelfunctionen. Theorie und Anwendungen. Reimer, Berlin (1878)

    Google Scholar 

  30. Henrici. P.: Applied and Computational Complex Analysis, vol. 3: Discrete Fourier Analysis—Cauchy Integrals—Construction of Conformal maps—Univalent Functions. John Wiley & Sons, New York (1986)

  31. Horowitz, D.: A further refinement for coefficient estimates of univalent functions. Proc. Amer. Math. Soc. 71, 217–221 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  32. Hummel, J. A.: A variational method for Gelfer functions. J. Analyse Math. 30, 271–280 (1976)

    MathSciNet  MATH  Google Scholar 

  33. Koebe, P.: Über die Uniformisierung beliebiger analytischer Kurven. Nachr. Kgl. Ges. Wiss. Göttingen, Math-Phys. Kl, pp. 191–210 (1907)

  34. Koebe, P.: Über die Unifomisierung der algebraischen Kurven durch automorphe Funktionen mit imaginärer Substitutionsgruppe. Nachr. Kgl. Ges. Wiss. Göttingen, Math-Phys. Kl, pp. 68–76 (1909)

  35. Koebe, P.: Über eine neue Methode der konformen Abbildung und Uniformisierung. Nachr. Kgl. Ges. Wiss. Göttingen, Math-Phys. Kl, pp. 844–848 (1912)

  36. Koepf, W.: On nonvanishing univalent functions with real coefficients. Math. Z. 192, 575–579 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  37. Koepf, W.: Extrempunkte und Stützpunkte in Familien nichtverschwindender schlichter Funktionen. Complex Variables 8, 153–171 (1987)

    MathSciNet  MATH  Google Scholar 

  38. Koepf, W.: Power series in Computer Algebra. J. Symb. Comp. 13, 581–603 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  39. Koepf, W.: Hypergeometric Summation. Vieweg, Braunschweig/Wiesbaden (1998)

    MATH  Google Scholar 

  40. Koepf, W.: Power series, Bieberbach conjecture and the de Branges and Weinstein functions. In: Sendra, J.R. (ed.) Proceedings of ISSAC 2003, pp. 169–175. Philadelphia, ACM, New York (2003).

  41. Koepf, W., Schmersau, D.: On the de Branges theorem. Complex Variables 31, 213–230 (1996)

    MathSciNet  MATH  Google Scholar 

  42. Laplace, P.-S.: Théorie des attractions des sphéroïdes et de la figure des planètes. Mémoires de l’Academie Royale des Sciences de Paris 113–196 (1782).

  43. Lebedev, N. A., Milin, I. M.: An inequality. Vestnik Leningrad Univ. 20, 157–158 (1965) (Russian)

    Google Scholar 

  44. Leeman, G. B.: The seventh coefficient of odd symmetric univalent functions. Duke Math. J. 43, 301–307 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  45. Legendre, A.-M.: Suite des recherches sur la figure des planètes. Mémoires de l’Academie Royale des Sciences de Paris, pp. 372–454 (1789).

  46. Leung, Y.: Successive coefficients of starlike functions. Bull. London Math. Soc. 10, 193–196 (1978)

    MathSciNet  MATH  Google Scholar 

  47. Leung, Y.: Robertson’s conjecture on the coefficients of close-to-convex functions. Proc. Amer. Math. Soc. 76, 89–94 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  48. Littlewood, J.E.: On inequalities in the theory of functions. Proc. London Math. Soc. (2) 23, 481–519 (1925)

    MathSciNet  Google Scholar 

  49. Littlewood, J.E. and Paley, R.E.A.C.: A proof that an odd schlicht function has bounded coefficients. J. London Math. Soc. 7, 167–169 (1932)

    MATH  Google Scholar 

  50. Löwner, K.: Untersuchungen über die Verzerrung bei konformen Abbildungen des Einheitskreises |z| < 1, die durch Funktionen mit nichtverschwindender Ableitung geliefert werden. S.-B. Verh. Sächs. Ges. Wiss. Leipzig 69, 89–106 (1917)

    Google Scholar 

  51. Löwner, K.: Untersuchungen über schlichte konforme Abbildungen des Einheitskreises I. Math. Ann. 89, 103–121 (1923)

    Article  MathSciNet  MATH  Google Scholar 

  52. Milin, I.M.: Estimation of coefficients of univalent functions. Dokl. Akad. Nauk SSSR 160 (1965), 769–771 (Russian) = Soviet Math. Dokl. 6, 196–198 (1965)

  53. Milin, I.M.: On the coefficients of univalent functions. Dokl. Akad. Nauk SSSR 176, 1015–1018 (1967) (Russian) = Soviet Math. Dokl. 8, 1255–1258 (1967)

    Google Scholar 

  54. Milin, I.M.: Univalent functions and orthonormal systems. Izdat. “Nauka”, Moskau, 1971 (Russian). English Translation: Amer. Math. Soc., Providence, R.I. (1977)

  55. Milin, I.M.: De Branges’ proof of the Bieberbach conjecture (Russian), Preprint (1984)

  56. Nehari, Z.: On the coefficients of Bieberbach-Eilenberg functions. J. Analyse Math. 23, 297–303 (1970).

    MathSciNet  MATH  Google Scholar 

  57. Nehari, Z.: A proof of |a 4| ≤ 4 by Loewner’s method. In: Clunie, J., Hayman, W.K. (ed.) Proceedings of the Symposium on Complex Analysis, Canterbury, 1973. London Math. Soc. Lecture Note Series, vol. 12, pp. 107–110. Cambridge University Press (1974).

  58. Nevanlinna, R.: Über die konforme Abbildung von Sterngebieten. Översikt av Finska Vetenskaps-Soc. Förh. 63(A), Nr. 6, 1–21 (1920–1921)

  59. Ozawa, M.: On the Bieberbach conjecture for the sixth coefficient. Kōdai Math. Sem. Rep. 21, 97–128 (1969)

    MathSciNet  MATH  Google Scholar 

  60. Ozawa, M.: An elementary proof of the Bieberbach conjecture for the sixth coefficient. Kōdai Math. Sem. Rep. 21, 129–132 (1969)

    MathSciNet  MATH  Google Scholar 

  61. Pederson, R.N.: A proof of the Bieberbach conjecture for the sixth coefficient. Arch. Rational. Mech. Anal. 31, 331–351 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  62. Pederson, R.N., Schiffer, M.: A proof of the Bieberbach conjecture for the fifth coefficient. Arch. Rational. Mech. Anal. 45, 161–193 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  63. Pommerenke, Ch.: Univalent functions. Vandenhoeck und Ruprecht. Göttingen-Zürich (1975)

  64. Pommerenke, Ch.: The Bieberbach Conjecture. Mathematical Intelligencer 7(2), 23–25 (1985)

    Article  MathSciNet  Google Scholar 

  65. Reade, M.O.: On close-to-convex univalent functions. Mich. Math. J. 3, 59–62 (1955)

    Article  MathSciNet  Google Scholar 

  66. Robertson, M.S.: On the theory of univalent functions. Ann. of Math. 37, 374–408 (1936)

    Article  MathSciNet  Google Scholar 

  67. Rogosinski, W.: Über positive harmonische Entwicklungen und typisch-reelle Potenzreihen. Math. Z. 35, 93–121 (1932)

    Article  MathSciNet  Google Scholar 

  68. Salvy, B., Zimmermann, P.: GFUN: A Maple package for the manipulation of generating and holonomic functions in one variable. ACM Transactions on Mathematical Software 20, 163–177 (1994)

    Article  MATH  Google Scholar 

  69. Schiffer, M.: A method of variation within the family of simple functions. Proc. London Math. Soc. 44, 432–449 (1938)

    MATH  Google Scholar 

  70. Study, E.: Vorlesungen über ausgewählte Gegenstände der Geometrie, 2. Heft: Konforme Abbildung einfach-zusammenhängender Bereiche. Teubner-Verlag, Leipzig-Berlin (1913)

    Google Scholar 

  71. Todorov, P.G.: A simple proof of the Bieberbach conjecture. Bull. Cl. Sci., VI. Sér, Acad. R. Belg. (3) 12, 335–356 (1992)

    Google Scholar 

  72. Weinstein, L.: The Bieberbach conjecture. Internat. Math. Res. Notices 5, 61–64 (1991)

    Article  MathSciNet  Google Scholar 

  73. Wilf, H.: A footnote on two proofs of the Bieberbach-de Branges Theorem. Bull. London Math. Soc. 26, 61–63 (1994)

    MathSciNet  MATH  Google Scholar 

  74. Wilf, H., Zeilberger, D.: An algorithmic proof theory for hypergeometric (ordinary and “q”) multisum/integral identities. Invent. Math. 103, 575–634 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  75. Zeilberger, D.: A fast algorithm for proving terminating hypergeometric identities. Discrete Math. 80, 207–211 (1990)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department of Mathematics, University of Kassel, Heinrich-Plett-Str. 40, 34132, Kassel, Germany

    Wolfram Koepf

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  1. Wolfram Koepf
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Correspondence to Wolfram Koepf.

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This article is dedicated to Dick Askey on occasion of his seventieth birthday.

2000 Mathematics Subject Classification Primary—30C50, 30C35, 30C45, 30C80, 33C20, 33C45, 33F10, 68W30

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Koepf, W. Bieberbach’s conjecture, the de Branges and Weinstein functions and the Askey-Gasper inequality. Ramanujan J 13, 103–129 (2007). https://doi.org/10.1007/s11139-006-0244-2

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