The Bieberbach Conjecture
and its Proof, Parts I & II
| Dr. John Marafino, February 5 and 12,
2007 |
Abstract: Let S denote the class of functions analytic and univalent in the unit disk, normalized by the conditions f(0)=0 and f'(0)=1. Then each f ε S has a Taylor series expansion of the form f(z) = z +
a_2 z^2 + a_3z^3 + ...., |z| <1.
A leading example of a function of
class S
is the Koebe
function
k(z) = z(1 -z )^{-2} = z + 2z^2
+ 3z^3 + ... .
In
1916 Bieberbach proved that if f ε S, then |a_2| ≤ 2, with
equality if and only if f is a rotation of the Koebe
function. With little to go on he then
conjectured that the coefficients of each function f ε S
satisfy |a_n| ≤ n for each n = 2,
3, …, and that strict inequality holds
for all n unless f is the Koebe function or one of its rotations.
Sixty-eight years later the conjecture had only been verified for n = 3, 4, 5, and 6. In 1984 Louis DeBranges was able to prove the full conjecture true. The theorem is now named after him. In our two-part colloquium series we will light-heartedly examine the mathematics and the characters surrounding the conjecture and its proof. |
|
|