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

Such functions are called schlicht [German for "natural, simple"]. The theorem then states that

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.

The normalizations

mean that

This can always be obtained by an affine transformation: starting with an arbitrary injective holomorphic function ${\displaystyle g}$ defined on the open unit disk and setting

Such functions ${\displaystyle g}$ are of interest because they appear in the Riemann mapping theorem.