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 and finally proven by . The statement concerns the Taylor coefficients 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 and . 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 for all , and it is schlicht, so we cannot find a stricter limit on the absolute value of the th coefficient. Schlicht functions The normalizations mean that This can always be obtained by an affine transformation: starting with an arbitrary injective holomorphic function defined on the open unit disk and setting Such functions are of interest because they appear in the Riemann mapping theorem. A schlicht function is defined as an analytic function that is one-to-one and satisfies and . A family of schlicht functions are the rotated Koebe functions with a complex number of absolute value . If is a schlicht function and for some , then is a rotated Koebe function. The condition of de Branges' theorem is not sufficient to show the function is schlicht, as the function shows: it is holomorphic on the unit disc and satisfies for all , but it is not injective since . History A survey of the history is given by Koepf (2007). proved , and stated the conjecture that . and independently proved the conjecture for starlike functions. Then Charles Loewner () proved , using the Löwner equation. His work was used by most later attempts, and is also applied in the theory of Schramm–Loewner evolution. proved that for all , showing that the Bieberbach conjecture is true up to a factor of Several authors later reduced the constant in the inequality below . If is a schlicht function then is an odd schlicht function. showed that its Taylor coefficients satisfy for all . They conjectured that can be replaced by as a natural generalization of the Bieberbach conjecture. The Littlewood–Paley conjecture