# Bieberbach conjecture

In

complex analysis, the

**Bieberbach conjecture** states a necessary condition on an analytic function to map the unit disk injectively to itself. The conjecture was proved in

1985 by

de Branges, with a proof that was subsequently much shortened by others.

The statement concerns the Taylor coefficients *a*_{n} of such a function, normalised as is always possible so that *a*_{0} is 0 and *a*_{1} is 1. It then states that |*a*_{n}| is at most *n*.