Editing De Branges's theorem (section)

{{Short description|Statement in complex analysis; formerly the Bieberbach conjecture}}
In [[complex analysis]], '''de Branges's theorem''', or the '''Bieberbach conjecture''', is a theorem that gives a [[necessary condition]] on a [[holomorphic function]] in order for it to map the [[unit disc|open unit disk]] of the [[complex plane]] [[injective]]ly to the complex plane. It was posed by {{harvs|txt|first=Ludwig |last=Bieberbach|authorlink=Ludwig Bieberbach|year=1916}} and finally proven by {{harvs|txt|authorlink=Louis de Branges de Bourcia|first=Louis |last=de Branges|year=1985}}.

The statement concerns the [[Taylor series|Taylor coefficient]]s <math>a_n</math> of a [[univalent function]], i.e. a one-to-one holomorphic function that maps the unit disk into the complex plane, normalized as is always possible so that <math>a_0=0</math> and <math>a_1=1</math>. That is, we consider a function defined on the open unit disk which is [[holomorphic function|holomorphic]]  and injective (''[[Univalent function|univalent]]'') with Taylor series of the form

:<math>f(z)=z+\sum_{n\geq 2} a_n z^n.</math>

Such functions are called ''schlicht''.  The theorem then states that

:<math> |a_n| \leq n \quad \text{for all }n\geq 2.</math>

The [[Koebe function]] (see below) is a function for which <math>a_n=n</math> for all <math>n</math>, and it is schlicht, so we cannot find a stricter limit on the absolute value of the <math>n</math>th coefficient.