de Branges, Louis A proof of the Bieberbach conjecture. (English) Zbl 0573.30014 Acta Math. 154, 137-152 (1985). 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 Cited in 26 ReviewsCited in 338 Documents MSC: 30C50 Coefficient problems for univalent and multivalent functions of one complex variable 30C55 General theory of univalent and multivalent functions of one complex variable Keywords:Bieberbach conjecture; Rogosinski conjecture; inequality of Lebedev- Milin Citations:Zbl 0014.40702; Zbl 0228.30011; Zbl 0355.33005; JFM 46.0552.01 PDFBibTeX XMLCite \textit{L. de Branges}, Acta Math. 154, 137--152 (1985; Zbl 0573.30014) Full Text: DOI Digital Library of Mathematical Functions: §16.23(iii) Conformal Mapping ‣ §16.23 Mathematical Applications ‣ Applications ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer 𝐺-Function Complex Function Theory ‣ §18.38(ii) Classical OP’s: Other Applications ‣ §18.38 Mathematical Applications ‣ Applications ‣ Chapter 18 Orthogonal Polynomials References: [1] Askey, R. &Gasper, G., Positive Jacobi sums II.Amer. J. 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Rational Mech. Anal., 45 (1972), 161–193. · Zbl 0241.30025 [15] Pommerenke, Ch., Über die Subordination analytischer Funktionen.J. Reine Angew. Math., 218 (1965), 159–173. · Zbl 0184.30601 [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 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.