## Démonstration de la conjecture de Bieberbach (d’après L. de Branges). (Demonstration of the Bieberbach conjecture (after L. de Branges)).(French)Zbl 0625.30019

Sémin. Bourbaki, 37e année, Vol. 1984/85, Exp. 649, Astérisque 133/134, 319-334 (1986).
[For the entire collection see Zbl 0577.00004.]
The author outlines the background to the Bieberbach conjecture [S.-B. Preuß. Akad. Wiss. Berlin 1916, 940-955 (1916)] that, if $$f(z)=z+a_ 2z^ 2+..$$. is analytic and one-one in $$\{| z| <1\}$$, then $$| a_ n| \leq n$$ (with equality only if $$f(z)=z(1-\lambda z)^ 2$$, $$| \lambda | =1)$$. He lists a series of other conjectures, stronger than the Bieberbach conjecture, due to Milin, Robertson, Rogosinski and Sheil-Small, all of which follow from the astonishing theorem of L. de Branges [Acta Math. 154, 137-152 (1985; Zbl 0573.30014)]: If $$f(z)=z+a_ 2z^ 2+..$$. is analytic and one-one in $$\{| z| <1\}$$ and if $$zf'(z)/f(z)=1+2\sum^{\infty}_{n=2}b_ nz^ n$$, then for each $$n\geq 0$$ we have $$\sum^{n}_{j=0}[(1/j)- 1/(n+1)](| b_ j|^ 2-1)\leq 0.$$
The author outlines the proof of this result, avoiding the use of hypergeometric functions.
Reviewer: D.A.Brannan

### 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

### Citations:

Zbl 0577.00004; Zbl 0573.30014
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