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On the computation of some Grunsky coefficients relevant to the Bieberbach conjecture


Author: George G. Ross
Journal: Math. Comp. 25 (1971), 733-741
MSC: Primary 30A34
DOI: https://doi.org/10.1090/S0025-5718-1971-0310227-5
MathSciNet review: 0310227
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Abstract: The Bieberbach coefficients $ {a_8},{a_{10}},{a_{12}}$ are parametrized in terms of the Grunsky coefficients. A digital computer is used to carry out the necessary integer calculations.


References [Enhancements On Off] (What's this?)

  • [1] L. Bieberbach, ``Ueber die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitkreises vermitteln,'' S.-B. Preuss. Akad. Wiss., v. 138, 1916. pp. 940-955.
  • [2] P. R. Garabedian, G. Ross & M. Schiffer, ``On the Bieberbach conjecture for even n,'' J. Math. Mech., v. 14, 1965, pp. 975-989. MR 32 #207. MR 0182725 (32:207)
  • [3] P. R. Garabedian & M. Schiffer, ``A proof of the Bieberbach conjecture for the fourth coefficient,'' J. Rational Mech. Anal., v. 4, 1955, pp. 427-465. MR 17, 24. MR 0070712 (17:24e)
  • [4] H. Grunsky, ``Koeffizientenbedingungen für schlicht abbildende meromorphe Funktionen,'' Math. Z., v. 45, 1939, pp. 29-61. MR 1545803
  • [5] J. A. Hummel, ``The Grunsky coefficients of schlicht functions,'' Proc. Amer. Math. Soc., v. 15, 1964, pp. 142-150. MR 28 #1287. MR 0158060 (28:1287)
  • [6] P. Koebe, ``Ueber die Uniformisierung der algebraischen Kurven. II,'' Math. Ann., v. 69, 1910, pp. 1-81. MR 1511577
  • [7] K. Loewner, ``Untersuchungen ueber schlichte konforme Abbildungen des Einheitskreises. I,'' Math. Ann., v. 89, 1923, pp. 103-121. MR 1512136
  • [8] R. N. Pederson, ``A proof of the Bieberbach conjecture for the sixth coefficient,'' Arch. Rational Mech. Anal., v. 31, 1968/69, pp. 331-351. MR 39 #431. MR 0239072 (39:431)
  • [9] M. Schiffer, ``A method of variation within the family of simple functions,'' Proc. London Math. Soc., v. 44, 1938, pp. 432-449.
  • [10] M. Schiffer, ``Variation of the Green function and theory of the p-valued functions,'' Amer. J. Math., v. 65, 1943, pp. 341-360. MR 4, 215. MR 0007925 (4:215c)
  • [11] A. C. Schaeffer & D. C. Spencer, ``The coefficients of schlicht functions,'' Duke Math. J., v. 10, 1943, pp. 611-635. MR 5, 175. MR 0009631 (5:175i)
  • [12] A. C. Schaeffer & D. C. Spencer, Coefficient Regions for schlicht Functions, Amer. Math. Soc. Colloq. Publ., vol. 35, Amer. Math. Soc., Providence, R.I., 1950. MR 12, 326. MR 0037908 (12:326c)
  • [13] I. Schur, ``Ein Satz ueber quadratische Formen mit komplexen Koeffizienten,'' Amer. J. Math., v. 67, 1945, pp. 472-480. MR 0014048 (7:234c)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1971-0310227-5
Keywords: Bieberbach conjecture, Koebe function, Grunsky coefficients, univalent functions, maximum principle, unitary matrix, Lagrange multipliers, overflow, fixed-point arithmetic
Article copyright: © Copyright 1971 American Mathematical Society

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