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Zbl 0573.30014
de Branges, Louis
A proof of the Bieberbach conjecture. (English)
[J] Acta Math. 154, 137-152 (1985). ISSN 0001-5962; ISSN 1871-2509/e

Let S denote the customary class of normalized univalent functions $$ f(z)=z+a\sb 2z\sp 2+...+a\sb nz\sp n+... $$ from the unit disk ${\bbfD}$ into ${\bbfC}$. When Bieberbach [Sitzungsber. Preuss. Akad. Wiss. 1916, 940--955 (1916; JFM 46.0552.01)] proved that $\vert a\sb 2\vert \le 2$ and equality only holds for the Koebe function $$ k(z)=z/(1-z)\sp 2=z+2z\sp 2+...+nz\sp n+... $$ and its rotations $e\sp{-i\alpha}k(e\sp{i\alpha}z)$, $\alpha\in {\bbfR}$, he conjectured that $\vert a\sb n\vert \le n$ for all n. In 1936 {\it M. S. Robertson} [Bull. Am. Math. Soc. 42, 366--370 (1936; Zbl 0014.40702)] conjectured that the odd functions $$ g(z)=z+b\sb 3z\sp 3+...+b\sb{2n-1}z\sp{2n-1}+... $$ of S satisfy the inequalities $$ 1+\vert b\sb 3\vert\sp 2+...+\vert b\sb{2n-1}\vert\sp 2\le n,\quad n=1,2,.... $$ This conjecture implies the Bieberbach inequalities and what is known as Rogosinski's conjecture saying that a function $h(z)=z+c\sb 2z\sp 2+...+c\sb nz\sp n+...$ which is holomorphic in ${\bbfD}$ and subordinate to a function of S satisfies the inequalities $\vert c\sb n\vert \le n$, $n=1,2,... $. The expansion $$ \log (f(z)/z)=\gamma\sb 1z+\gamma\sb 2z\sp 2+...+\gamma\sb nz\sp n+... $$ defines the logarithmic coefficients $\gamma\sb n$ of a function in S. {\it I. M. Milin} ["Univalent functions and orthonormal systems" (1971; Zbl 0228.30011)] conjectured that $$ (*)\quad \sum\sp{n}\sb{k=1}(1- k/(n+1))(k\vert \gamma\sb k\vert\sp 2-(1/k))\le 0 $$ 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. \par Now, in the early spring of 1984, {\it 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 $\sigma\sb n(t)$ and the related theory of square summable power series, and a theorem of {\it R. Askey} and {\it G. Gasper} [Am. J. Math. 98, 709--737 (1976; Zbl 0355.33005)] on positive sums of Jacobi polynomials.
[A.Pfluger]

MSC 2000:: *30C50 Coefficient problems for univalent and multivalent functions
30C55 General theory of univalent and multivalent functions

Keywords: Bieberbach conjecture; Rogosinski conjecture; inequality of Lebedev- Milin

Citations: Zbl 0014.40702; Zbl 0228.30011; Zbl 0355.33005; JFM 46.0552.01

Cited in: Zbl 1186.30001 Zbl 1088.30010 Zbl 1062.47002 Zbl 1003.30003 Zbl 1096.30502 Zbl 0920.33001 Zbl 0835.30009 Zbl 0862.30019 Zbl 0786.30012 Zbl 0848.46011 Zbl 0806.30017 Zbl 0743.30021 Zbl 0731.30006 Zbl 0729.30002 Zbl 0644.30010 Zbl 0624.30024 Zbl 1056.47503 Zbl 0625.30019 Zbl 0619.30018 Zbl 0608.30021 Zbl 0605.30018 Zbl 0553.30012

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