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 and finally proven by.
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 with Taylor series of the form
Such functions are called schlicht. The theorem then states that
The Koebe function 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.

Schlicht functions

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 and f ' = 1. A family of schlicht functions are the rotated Koebe functions
with α a complex number of absolute value 1. If f is a schlicht function and |an| = n for some n ≥ 2, then f 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 |an|≤n for all n, but it is not injective since f = f.

History

A survey of the history is given by .
proved |a2| ≤ 2, and stated the conjecture that |an| ≤ n. and independently proved the conjecture for starlike functions.
Then Charles Loewner proved |a3| ≤ 3, 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 |an| ≤ en for all n, showing that the Bieberbach conjecture is true up to a factor of e = 2.718... Several authors later reduced the constant in the inequality below e.
If f = z +... is a schlicht function then φ = f1/2 is an odd schlicht function.
showed that its Taylor coefficients satisfy bk ≤ 14 for all k. They conjectured that 14 can be replaced by 1 as a natural generalization of the Bieberbach conjecture. The Littlewood–Paley conjecture easily implies the Bieberbach conjecture using the Cauchy inequality, but it was soon disproved by, who showed there is an odd schlicht function with b5 = 1/2 + exp = 1.013..., and that this is the maximum possible value of b5. Isaak Milin later showed that 14 can be replaced by 1.14, and Hayman showed that the numbers bk have a limit less than 1 if f is not a Koebe function. So the limit is always less than or equal to 1, 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.
The Robertson conjecture states that if
is an odd schlicht function in the unit disk with b1=1 then for all positive integers n,
Robertson observed that his conjecture is still strong enough to imply the Bieberbach conjecture, and proved it for n = 3. 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 n, in particular proved |a4| ≤ 4, and proved |a6| ≤ 6, and proved |a5| ≤ 5.
proved that the limit of an/n exists, and has absolute value less than 1 unless f is a Koebe function. In particular this showed that for any f 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 n,
where the logarithmic coefficients γn of f are given by

showed using the Lebedev–Milin inequality that the Milin conjecture implies the Robertson conjecture and therefore the Bieberbach conjecture.
Finally proved |an| ≤ n for all n.

de Branges's proof

The proof uses a type of Hilbert spaces of entire functions. The study of these spaces grew into a sub-field of complex analysis and the spaces have come to be called de Branges spaces. De Branges proved the stronger Milin conjecture on logarithmic coefficients. This was already known to imply the Robertson conjecture about odd univalent functions, which in turn was known to imply the Bieberbach conjecture about schlicht functions. 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 and then asked Richard Askey whether he knew of any similar inequalities. Askey pointed out that 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 when de Branges visited in 1984.
De Branges proved the following result, which for ν = 0 implies the Milin conjecture.
Suppose that ν > −3/2 and σn are real numbers for positive integers n with limit 0 and such that
is non-negative, non-increasing, and has limit 0. Then for all Riemann mapping functions F = z + ... univalent in the unit disk with
the maximinum value of
is achieved by the Koebe function z/2.
A simplified version of the proof was published in 1985 by Carl FitzGerald and Christian Pommerenke, and an even shorter description by Jacob Korevaar.