De Branges's theorem

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 open unit disk of the complex plane injectively to the complex plane. It was posed by Ludwig Bieberbach (1916) and finally proven by Louis de Branges (1985).

The statement concerns the Taylor coefficients ${\displaystyle a_{n}}$ 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 ${\displaystyle a_{0}=0}$ and ${\displaystyle a_{1}=1}$. That is, we consider a function defined on the open unit disk which is holomorphic and injective (univalent) with Taylor series of the form

${\displaystyle f(z)=z+\sum _{n\geq 2}a_{n}z^{n}.}$

Such functions are called schlicht. The theorem then states that

${\displaystyle |a_{n}|\leq n\quad {\text{for all }}n\geq 2.}$

The Koebe function (see below) is a function for which ${\displaystyle a_{n}=n}$ for all ${\displaystyle n}$, and it is schlicht, so we cannot find a stricter limit on the absolute value of the ${\displaystyle n}$th coefficient.