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).
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
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.
, and it is schlicht, so we cannot find a stricter limit on the absolute value of the
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
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
Löwner (1917) and Nevanlinna (1921) independently proved the conjecture for starlike functions.
His work was used by most later attempts, and is also applied in the theory of Schramm–Loewner evolution.
, showing that the Bieberbach conjecture is true up to a factor of
Paley and Littlewood (1932) showed that its Taylor coefficients satisfy
The Littlewood–Paley conjecture easily implies the Bieberbach conjecture using the Cauchy inequality, but it was soon disproved by Fekete & Szegő (1933), who showed there is an odd schlicht function with
, meaning that Littlewood and Paley's conjecture is true for all but a finite number of coefficients.
A weaker form of Littlewood and Paley's conjecture was found by Robertson (1936).
The Robertson conjecture states that if is an odd schlicht function in the unit disk with
This conjecture introduced the key idea of bounding various quadratic functions of the coefficients rather than the coefficients themselves, which is equivalent to bounding norms of elements in certain Hilbert spaces of schlicht functions.
There were several proofs of the Bieberbach conjecture for certain higher values of
there can be at most a finite number of exceptions to the Bieberbach conjecture.
The Milin conjecture states that for each schlicht function on the unit disk, and for all positive integers
The proof uses a type of Hilbert space of entire functions.
This was already known to imply the Robertson conjecture (Robertson 1936) about odd univalent functions, which in turn was known to imply the Bieberbach conjecture about schlicht functions (Bieberbach 1916).
His proof uses the Loewner equation, the Askey–Gasper inequality about Jacobi polynomials, and the Lebedev–Milin inequality on exponentiated power series.
De Branges reduced the conjecture to some inequalities for Jacobi polynomials, and verified the first few by hand.
Walter Gautschi verified more of these inequalities by computer for de Branges (proving the Bieberbach conjecture for the first 30 or so coefficients) and then asked Richard Askey whether he knew of any similar inequalities.
Askey pointed out that Askey & Gasper (1976) had proved the necessary inequalities eight years before, which allowed de Branges to complete his proof.
The first version was very long and had some minor mistakes which caused some skepticism about it, but these were corrected with the help of members of the Leningrad seminar on Geometric Function Theory (Leningrad Department of Steklov Mathematical Institute) when de Branges visited in 1984.
univalent in the unit disk with the maximum value of is achieved by the Koebe function
A simplified version of the proof was published in 1985 by Carl FitzGerald and Christian Pommerenke (FitzGerald & Pommerenke (1985)), and an even shorter description by Jacob Korevaar (Korevaar (1986)).
A very short proof avoiding use of the inequalities of Askey and Gasper was later found by Lenard Weinstein (Weinstein (1991)).