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 an of a univalent function, i.e. a one-to-one holomorphic function that maps the unit disk into the complex plane, normalized as is always possible so that a0 = 0 and a1 = 1. That is, we consider a function defined on the open unit disk which is holomorphic and injective (univalent) with Taylor series of the form
such functions are called schlicht. The theorem then states that
The Koebe function (see below) is a function in which an = n for all n, and it is schlicht, so we cannot find a stricter limit on the absolute value of the nth coefficient.
The normalizations
mean that
This can always be obtained by an affine transformation: starting with an arbitrary injective holomorphic function g defined on the open unit disk and setting
Such functions g are of interest because they appear in the Riemann mapping theorem.
A schlicht function is defined as an analytic function f that is one-to-one and satisfies f(0) = 0 and f '(0) = 1. A family of schlicht functions are the rotated Koebe functions
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