De Branges' theorem
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In complex analysis, de Branges' theorem, named after Louis de Branges, formerly called the Bieberbach conjecture, after Ludwig Bieberbach, 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 an of such a function, normalised as is always possible so that a0 is 0 and a1 is 1. That is, we consider
- <math>f(z)=z+\sum_{n\geq 2} a_n z^n.<math>
The theorem states that
- <math>\left| a_n \right| \leq n.\,<math>
The normalisation a0 says that f(0) = 0; this can be assured by composing f with a Möbius 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.de:Bieberbachsche Vermutung he:השערת ביברבך