De Branges' theorem

In complex analysis, de Branges' theorem, formerly the Bieberbach conjecture, states a necessary condition on an analytic function to map the unit disk injectively to the complex plane. The conjecture was stated in 1916 by Bieberbach but proved only 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 a₀ is 0 and a₁ is 1. That is, we consider

f(z) = Σ a_nzⁿ.

It then states that

|a_n| ≤ n.

The normalisation a₀ is to say that we may assume f(0) = 0; this can be assured by composing f with a Moebius transformation that preserves the unit circle.

The case n = 1 here is known as the Schwarz lemma, and was known in the nineteenth century. It is a consequence of the maximum modulus principle, applied to

f(z)/z.

For n ≥ 2 numerous partial results were known before 1985.