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Bieberbach's conjecture (Theorem)

The following theorem is known as the Bieberbach conjecture, even though it has now been proven. Bieberbach proposed it in 1916 and it was finally proven in 1984 by Louis de Branges.

Firstly note that if $ f \colon {\mathbb{D}} \to {\mathbb{C}}$ is a schlicht function (univalent, $ f(0) = 0$ and $ f'(0) = 1$) then $ f$ has a power series representation as

$\displaystyle f(z) = z + a_2 z^2 + a_3 z^3 + \cdots = z + \sum_{k=2}^\infty a_k z^k .$    

Theorem 1 (Bieberbach)   Suppose that $ f$ is a schlicht function, then $ \lvert a_k \rvert \leq k$ for all $ k \geq 2$ and furthermore if there is some integer $ k$ such that $ \lvert a_k \rvert = k$, then $ f$ is some rotation of the Koebe function.

In fact if $ f$ is a rotation of the Koebe function then $ \lvert a_k \rvert = k$ for all $ k$.

Bibliography

1
John B. Conway. Functions of One Complex Variable II. Springer-Verlag, New York, New York, 1995.



"Bieberbach's conjecture" is owned by jirka.
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Other names:  Bieberbach conjecture
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Cross-references: rotation of the Koebe function, integer, representation, power series, univalent, schlicht function, even
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This is version 4 of Bieberbach's conjecture, born on 2004-06-07, modified 2006-09-17.
Object id is 5899, canonical name is BieberbachsConjecture.
Accessed 5222 times total.

Classification:
AMS MSC: 30C45 (Functions of a complex variable :: Geometric function theory :: Special classes of univalent and multivalent functions )
 30C55 (Functions of a complex variable :: Geometric function theory :: General theory of univalent and multivalent functions)
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