A proof of the Bieberbach conjecture
type: Item
instance of: scholarly article
title: A proof of the Bieberbach conjecture
zbMATH Open document ID: 0573.30014
DOI: 10.1007/BF02392821
published in: Acta Mathematica
publication date: 1985-01-01T00:00:00Z
review text: 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 \textit{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. \textit{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, \textit{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 fu…
Mathematics Subject Classification ID: 30C50
Mathematics Subject Classification ID: 30C55
zbMATH DE Number: 3914658
zbMATH Keywords: Bieberbach conjecture
zbMATH Keywords: Rogosinski conjecture
zbMATH Keywords: inequality of Lebedev- Milin
Wikidata QID: Q55895520
MaRDI profile type: Publication
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author: Louis de Branges de Bourcia
description: scientific article; zbMATH DE number 3914658