zbMATH — the first resource for mathematics

A proof of the Bieberbach conjecture. (English) Zbl 0573.30014
Let S denote the customary class of normalized univalent functions \[ f(z)=z+a_ 2z^ 2+...+a_ nz^ n+... \] from the unit disk \({\mathbb{D}}\) into \({\mathbb{C}}\). When Bieberbach [Sitzungsber. Preuß. Akad. Wiss. 1916, 940–955 (1916; JFM 46.0552.01)] proved that \(| a_ 2| \leq 2\) and equality only holds for the Koebe function \[ k(z)=z/(1-z)^ 2=z+2z^ 2+...+nz^ n+... \] and its rotations \(e^{-i\alpha}k(e^{i\alpha}z)\), \(\alpha\in {\mathbb{R}}\), he conjectured that \(| a_ n| \leq n\) for all n. In 1936 M. S. Robertson [Bull. Am. Math. Soc. 42, 366–370 (1936; Zbl 0014.40702)] conjectured that the odd functions \[ g(z)=z+b_ 3z^ 3+...+b_{2n-1}z^{2n-1}+... \] of S satisfy the inequalities \[ 1+| b_ 3|^ 2+...+| b_{2n-1}|^ 2\leq n,\quad n=1,2,.... \] This conjecture implies the Bieberbach inequalities and what is known as Rogosinski’s conjecture saying that a function \(h(z)=z+c_ 2z^ 2+...+c_ nz^ n+...\) which is holomorphic in \({\mathbb{D}}\) and subordinate to a function of S satisfies the inequalities \(| c_ n| \leq n\), \(n=1,2,... \). The expansion \[ \log (f(z)/z)=\gamma_ 1z+\gamma_ 2z^ 2+...+\gamma_ nz^ n+... \] defines the logarithmic coefficients \(\gamma_ n\) of a function in S. I. M. Milin [”Univalent functions and orthonormal systems” (1971; Zbl 0228.30011)] conjectured that \[ (*)\quad \sum^{n}_{k=1}(1- k/(n+1))(k| \gamma_ k|^ 2-(1/k))\leq 0 \] for all \(n=1,2,..\). and all f in S. By an inequality of Lebedev-Milin this implies Robertson’s, hence Rogosinski’s and Bieberbach’s conjecture.
Now, in the early spring of 1984, L. de Branges [Preprint E-5-84, Leningrad Branch of the V. A. Steklov Mathematical Institute (1984)] proved Milin’s conjecture to hold true and the equality sign in (*) to appear only for Koebe functions. By this the four problems mentioned above were settled at once. The proof presented in this paper is based on the theory of Loewner chains, the de Branges’ system of differential equations for the weight functions \(\sigma_ n(t)\) and the related theory of square summable power series, and a theorem of R. Askey and G. Gasper [Am. J. Math. 98, 709–737 (1976; Zbl 0355.33005)] on positive sums of Jacobi polynomials.
Reviewer: A.Pfluger

30C50 Coefficient problems for univalent and multivalent functions of one complex variable
30C55 General theory of univalent and multivalent functions of one complex variable
Full Text: DOI
[1] Askey, R. &Gasper, G., Positive Jacobi sums II.Amer. J. Math., 98 (1976), 709–737. · Zbl 0355.33005 · doi:10.2307/2373813
[2] Bieberbach, L., Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln.Sitzungsberichte Preussische Akademie der Wissenschaften, 1916, 940–955. · JFM 46.0552.01
[3] De Branges, L., Coefficient estimates.J. Math. Anal. Appl., 82 (1981), 420–450. · Zbl 0494.30017 · doi:10.1016/0022-247X(81)90207-9
[4] –, Grunsky spaces of analytic functions.Bull. Sci. Math., 105 (1981), 401–406. · Zbl 0496.30009
[5] –, Löwner expansions.J. Math. Anal. Appl., 100 (1984), 323–337. · Zbl 0552.30011 · doi:10.1016/0022-247X(84)90084-2
[6] De Branges, L.,A proof of the Bieberbach conjecture. Preprint E-5-84, Leningrad Branch of the V. A. Steklov Mathematical Institute, 1984. · Zbl 0573.30014
[7] Garabedian, P. R. &Schiffer, M., A proof of the Bieberbach conjecture for the fourth coefficient.Arch. Rational Mech. Anal., 4 (1955), 427–455. · Zbl 0065.06902
[8] Grinšpan, A. Ž., Logarithmic coefficients of functions of the class.S. Sibirsk. Mat. Ž., 13 (1972), 1145–1151 (Russian).Siberian Math. J., 13 (1972), 793–801 (English).
[9] Löwner, K., Untersuchungen über schlichte konforme Abbildungen des Einheitskreises.Math. Ann., 89 (1923), 102–121. · JFM 49.0714.01
[10] Milin, I. M., On the coefficients of univalent functions.Dokl. Akad. Nauk SSSR, 176 (1967), 1015–1018 (Russian).Soviet Math. Dokl. 8 (1967), 1255–1258 (English). · Zbl 0176.03201
[11] –,Univalent functions and orthonormal systems. Nauka, Moscow, 1971 (Russian). Translations of Mathematical Mongraphs, volume 49. American Mathematical Society, Providence, 1977.
[12] Ozawa, M., An elementary proof of the Bieberbach conjecture for the sixth coefficient.Kodai Math. Sem. Rep., 21 (1969), 129–132. · Zbl 0202.07201 · doi:10.2996/kmj/1138845875
[13] Pederson, R. N., A proof of the Bieberbach conjecture for the sixth coefficient.Arch. Rational Mech. Anal., 31 (1968), 331–351. · Zbl 0184.10501 · doi:10.1007/BF00251415
[14] Pederson, R. &Schiffer, M., A proof of the Bieberbach conjecture of the fifth coefficient.Arch. Rational Mech. Anal., 45 (1972), 161–193. · Zbl 0241.30025 · doi:10.1007/BF00281531
[15] Pommerenke, Ch., Über die Subordination analytischer Funktionen.J. Reine Angew. Math., 218 (1965), 159–173. · Zbl 0184.30601 · doi:10.1515/crll.1965.218.159
[16] –,Univalent functions. Vandenhoeck & Ruprecht, Göttingen, 1975. · Zbl 0306.30018
[17] Robertson, M. S., A remark on the odd schlicht functions.Bull. Amer. Math. Soc., 42 (1936), 366–370. · Zbl 0014.40702 · doi:10.1090/S0002-9904-1936-06300-7
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.