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A proof of the Bieberbach conjecture. (English) Zbl 0573.30014

Let S denote the customary class of normalized univalent functions

$f\left(z\right)=z+{a}_{2}{z}^{2}+···+{a}_{n}{z}^{n}+···$

from the unit disk $𝔻$ into $ℂ$. When Bieberbach [Sitzungsber. Preuß. Akad. Wiss. 1916, 940–955 (1916; JFM 46.0552.01)] proved that $|{a}_{2}|\le 2$ and equality only holds for the Koebe function

$k\left(z\right)=z/{\left(1-z\right)}^{2}=z+2{z}^{2}+···+n{z}^{n}+···$

and its rotations ${e}^{-i\alpha }k\left({e}^{i\alpha }z\right)$, $\alpha \in ℝ$, he conjectured that $|{a}_{n}|\le n$ for all n. In 1936 M. S. Robertson [Bull. Am. Math. Soc. 42, 366–370 (1936; Zbl 0014.40702)] conjectured that the odd functions

$g\left(z\right)=z+{b}_{3}{z}^{3}+···+{b}_{2n-1}{z}^{2n-1}+···$

of S satisfy the inequalities

$1+|{b}_{3}{|}^{2}+···+{|{b}_{2n-1}|}^{2}\le n,\phantom{\rule{1.em}{0ex}}n=1,2,····$

This conjecture implies the Bieberbach inequalities and what is known as Rogosinski’s conjecture saying that a function $h\left(z\right)=z+{c}_{2}{z}^{2}+···+{c}_{n}{z}^{n}+···$ which is holomorphic in $𝔻$ and subordinate to a function of S satisfies the inequalities $|{c}_{n}|\le n$, $n=1,2,···$. The expansion

$log\left(f\left(z\right)/z\right)={\gamma }_{1}z+{\gamma }_{2}{z}^{2}+···+{\gamma }_{n}{z}^{n}+···$

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

$\left(*\right)\phantom{\rule{1.em}{0ex}}\sum _{k=1}^{n}\left(1-k/\left(n+1\right)\right)\left(k|{\gamma }_{k}{|}^{2}-\left(1/k\right)\right)\le 0$

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.

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 ${\sigma }_{n}\left(t\right)$ 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

##### MSC:
 30C50 Coefficient problems for univalent and multivalent functions 30C55 General theory of univalent and multivalent functions
##### References:
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