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 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 was proposed by Ludwig Bieberbach in $1916$.
After attempts by many mathematicians, it was finally proved true by Louis de Branges in $1984$.