Bieberbach’s conjecture

The following theorem is known as the Bieberbach conjectureMathworldPlanetmath, 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:𝔻 is a schlicht functionMathworldPlanetmath (univalent, f(0)=0 and f(0)=1) then f has a power seriesMathworldPlanetmath representation as

Theorem (Bieberbach).

Suppose that f is a schlicht function, then |ak|k for all k2 and furthermore if there is some integer k such that |ak|=k, then f is some rotation of the Koebe function.

In fact if f is a rotation of the Koebe function then |ak|=k for all k.


  • 1 John B. Conway. . Springer-Verlag, New York, New York, 1995.
Title Bieberbach’s conjecture
Canonical name BieberbachsConjecture
Date of creation 2013-03-22 14:24:07
Last modified on 2013-03-22 14:24:07
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 7
Author jirka (4157)
Entry type Theorem
Classification msc 30C55
Classification msc 30C45
Synonym Bieberbach conjecture