Bieberbach Conjecture

Theorem

Let $f$ be a holomorphic complex function defined as:

$\forall z \in \C: \map f z = z + a_2 z^2 + a_3 z^3 + \cdots$

where the $a_n$ are complex.

Let $f$ be injective for $\size z < 1$.

Then:

$\forall n \ge 2: \size {a_n} \le n$

Proof

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Also known as

The Bieberbach Conjecture is also known as de Branges's Theorem, for Louis de Branges who proved it.

Source of Name

This entry was named for Ludwig Georg Elias Moses Bieberbach.

Historical Note

The Bieberbach Conjecture was proposed by Ludwig Bieberbach in $1916$.

After attempts by many mathematicians, it was finally proved true by Louis de Branges in $1984$.

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Bieberbach conjecture