De Branges's Theorem

De Branges's Theorem

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 a_n of such a function, normalized as is always possible so that a₀ = 0 and a₁ = 1. That is, we consider a holomorphic function of the form

which is defined and injective on the open unit disk (such functions are also called univalent or schlicht functions). The theorem then states that