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Ross, George G.

On the computation of some Grunsky coefficients relevant to the Bieberbach conjecture. (English) Zbl 0233.30009

Math. Comput. 25, 733-741 (1971).
Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page
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MSC:

30C50 Coefficient problems for univalent and multivalent functions of one complex variable
30C75 Extremal problems for conformal and quasiconformal mappings, other methods
30B10 Power series (including lacunary series) in one complex variable
30-04 Software, source code, etc. for problems pertaining to functions of a complex variable
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References:

[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. · JFM 46.0552.01
[2] P. R. Garabedian, G. G. Ross, and M. M. Schiffer, On the Bieberbach conjecture for even \?, J. Math. Mech. 14 (1965), 975 – 989. · Zbl 0141.26901
[3] P. R. Garabedian and M. Schiffer, A proof of the Bieberbach conjecture for the fourth coefficient, J. Rational Mech. Anal. 4 (1955), 427 – 465. · Zbl 0065.06902
[4] Helmut Grunsky, Koeffizientenbedingungen für schlicht abbildende meromorphe Funktionen, Math. Z. 45 (1939), no. 1, 29 – 61 (German). · Zbl 0022.15103 · doi:10.1007/BF01580272
[5] J. A. Hummel, The Grunsky coefficients of a schlicht function, Proc. Amer. Math. Soc. 15 (1964), 142 – 150. · Zbl 0137.05302
[6] Paul Koebe, Über die Uniformisierung der algebraischen Kurven. II, Math. Ann. 69 (1910), no. 1, 1 – 81 (German). · JFM 41.0480.02 · doi:10.1007/BF01455152
[7] Karl Löwner, Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I, Math. Ann. 89 (1923), no. 1-2, 103 – 121 (German). · JFM 49.0714.01 · doi:10.1007/BF01448091
[8] Roger N. Pederson, A proof of the Bieberbach conjecture for the sixth coefficient, Arch. Rational Mech. Anal. 31 (1968/1969), 331 – 351. · Zbl 0184.10501 · doi:10.1007/BF00251415
[9] M. Schiffer, “A method of variation within the family of simple functions,” Proc. London Math. Soc., v. 44, 1938, pp. 432-449. · Zbl 0019.22201
[10] Menahem Schiffer, Variation of the Green function and theory of the \?-valued functions, Amer. J. Math. 65 (1943), 341 – 360. · Zbl 0060.23705 · doi:10.2307/2371820
[11] A. C. Schaeffer and D. C. Spencer, The coefficients of schlicht functions, Duke Math. J. 10 (1943), 611 – 635. · Zbl 0060.20404
[12] A. C. Schaeffer and D. C. Spencer, Coefficient Regions for Schlicht Functions, American Mathematical Society Colloquium Publications, Vol. 35, American Mathematical Society, New York, N. Y., 1950. With a Chapter on the Region of the Derivative of a Schlicht Function by Arthur Grad. · Zbl 0049.06003
[13] Issai Schur, Ein Satz ueber quadratische Formen mit komplexen Koeffizienten, Amer. J. Math. 67 (1945), 472 – 480 (German). · Zbl 0060.28007 · doi:10.2307/2371974
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