The de Branges theorem on univalent functions
HTML articles powered by AMS MathViewer
- by Carl H. FitzGerald and Ch. Pommerenke
- Trans. Amer. Math. Soc. 290 (1985), 683-690
- DOI: https://doi.org/10.1090/S0002-9947-1985-0792819-9
- PDF | Request permission
Abstract:
We present a simplified version of the de Branges proof of the Lebedev-Milin conjecture which implies the Robertson and Bieberbach conjectures. As an application of an analysis of the technique, it is shown that the method could not be used directly to prove the Bieberbach conjecture.References
- Richard Askey and George Gasper, Positive Jacobi polynomial sums. II, Amer. J. Math. 98 (1976), no. 3, 709–737. MR 430358, DOI 10.2307/2373813 L. Bieberbach, Über die Koeffizienten derjenigan Potenzreichen, welche eine schlichte Abbildung des Einheitskreisses vermitteln, S.-B. Preuse. Akad. Wiss., Phys.-Math. Kl. (1916), 940-955. L. de Branges, A proof of the Bieberbach conjecture, Preprint E-5-84, Steklov Math. Institute, LOMI, Leningrad, 1984, pp. 1-21. —, A proof of the Bieberbach conjecture (to appear). —, Square summable power series, 2nd ed. (to appear).
- Peter L. Duren, Univalent functions, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. MR 708494
- George Gasper, A short proof of an inequality used by de Branges in his proof of the Bieberbach, Robertson and Milin conjectures, Complex Variables Theory Appl. 7 (1986), no. 1-3, 45–50. MR 877650, DOI 10.1080/17476938608814185
- Karl Löwner, Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I, Math. Ann. 89 (1923), no. 1-2, 103–121 (German). MR 1512136, DOI 10.1007/BF01448091
- I. M. Milin, The coefficients of schlicht functions, Dokl. Akad. Nauk SSSR 176 (1967), 1015–1018 (Russian). MR 0222276
- I. M. Milin, Univalent functions and orthonormal systems, Translations of Mathematical Monographs, Vol. 49, American Mathematical Society, Providence, R.I., 1977. Translated from the Russian. MR 0427620
- Christian Pommerenke, Univalent functions, Studia Mathematica/Mathematische Lehrbücher, Band XXV, Vandenhoeck & Ruprecht, Göttingen, 1975. With a chapter on quadratic differentials by Gerd Jensen. MR 0507768
- M. S. Robertson, A remark on the odd schlicht functions, Bull. Amer. Math. Soc. 42 (1936), no. 6, 366–370. MR 1563301, DOI 10.1090/S0002-9904-1936-06300-7
- M. S. Robertson, Quasi-subordination and coefficient conjectures, Bull. Amer. Math. Soc. 76 (1970), 1–9. MR 251210, DOI 10.1090/S0002-9904-1970-12356-4 G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1959.
Bibliographic Information
- © Copyright 1985 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 290 (1985), 683-690
- MSC: Primary 30C50
- DOI: https://doi.org/10.1090/S0002-9947-1985-0792819-9
- MathSciNet review: 792819