# De Branges's theorem

Template:Lowercase title
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 Template:Harvs and finally proven by Template:Harvs.

The statement concerns the Taylor coefficients *a _{n}* of such a function, normalized as is always possible so that

*a*

_{0}= 0 and

*a*

_{1}= 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

## Schlicht functions

The normalizations

*a*_{0}= 0 and*a*_{1}= 1

mean that

*f*(0) = 0 and*f*'(0) = 1;

this can always be assured by a linear fractional 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

with α a complex number of absolute value 1. If *f* is a schlicht function and |*a*_{n}| = *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 |*a*_{n}|≤*n* for all *n*, but it is not injective since *f*(−1/2 + *z*) = *f*(−1/2 − *z*).

## History

A survey of the history is given by Koepf (2007).

Template:Harvtxt proved |*a*_{2}| ≤ 2, and stated the conjecture that |*a*_{n}| ≤ *n*. Template:Harvtxt and Template:Harvtxt independently proved the conjecture for starlike functions.
Then Charles Loewner (Template:Harvtxt) proved |*a*_{3}| ≤ 3, using the Löwner equation. His work was used by most later attempts, and is also applied in the theory of Schramm–Loewner evolution.

Template:Harvtxt proved that |*a*_{n}| ≤ *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*) = *z* + ... is a schlicht function then φ(*z*) = *f*(*z*^{2})^{1/2} is an odd schlicht function.
Template:Harvs showed that its Taylor coefficients satisfy *b*_{k} ≤ 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 Template:Harvtxt, who showed there is an odd schlicht function with *b*_{5} = 1/2 + exp(−2/3) = 1.013..., and that this is the maximum possible value of *b*_{5}. (Milin later showed that 14 can be replaced by 1.14., and Hayman showed that the numbers *b*_{k} have a limit less than 1 if φ is not a Koebe function, so Littlewood and Paley's conjecture is true for all but a finite number of coefficients of any function.) A weaker form of Littlewood and Paley's conjecture was found by Template:Harvtxt.

The **Robertson conjecture** states that if

is an odd schlicht function in the unit disk with *b*_{1}=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 Template:Harvtxt proved |*a*_{4}| ≤ 4, Template:Harvtxt and Template:Harvtxt proved |*a*_{6}| ≤ 6, and Template:Harvtxt proved |*a*_{5}| ≤ 5.

Template:Harvtxt proved that the limit of *a*_{n}/*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 simple function on the unit disk, and for all positive integers *n*,

where the **logarithmic coefficients** γ_{n} of *f* are given by

Template:Harvtxt showed using the Lebedev–Milin inequality that the Milin conjecture (later proved by de Branges) implies the Robertson conjecture and therefore the Bieberbach conjecture.

Finally Template:Harvtxt proved |*a*_{n}| ≤ *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 come to be called de Branges spaces and the functions de Branges functions. De Branges proved the stronger Milin conjecture Template:Harv on logarithmic coefficients. This was already known to imply the Robertson conjecture Template:Harv about odd univalent functions, which in turn was known to imply the Bieberbach conjecture about simple functions Template:Harv. 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 if he knew of any similar inequalities. Askey pointed out that Template:Harvtxt 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.

De Branges proved the following result, which for ν = 0 implies the Milin conjecture (and therefore the Bieberbach conjecture).
Suppose that ν > −3/2 and σ_{n} 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*) = *z* + ... univalent in the unit disk with

the maximinum value of

is achieved by the Koebe function *z*/(1 − *z*)^{2}.

## References

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}.

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- Koepf, Wolfram (2007),
*Bieberbach’s Conjecture, the de Branges and Weinstein Functions and the Askey-Gasper Inequality* - {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }} (Translation of the 1971 Russian edition)

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}

- {{#invoke:citation/CS1|citation

|CitationClass=citation }}