Bieberbach conjecture

In complex analysis, the Bierberbach 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 a₀is 0 and a₁ is 1. It then states that |a_n| is at most n.