De Branges's theorem#Schlicht functions

The normalizations

:$a\_0=0\; \backslash \; \backslash text\{and\}\backslash \; a\_1=1$

mean that

:$f(0)=0\backslash \; \backslash text\{and\}\backslash \; f\text{'}(0)=1\; .$

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

:$f(z)=\backslash frac\{g(z)-g(0)\}\{g\text{'}(0)\}.$

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 $f\text{'}(0)=1$. A family of schlicht functions are the rotated Koebe functions

:$f\_\backslash alpha(z)=\backslash frac\{z\}\{(1-\backslash alpha\; z)^2\}=\backslash sum\_\{n=1\}^\backslash infty\; n\backslash alpha^\{n-1\}\; z^n$

with $\backslash alpha$ a complex number of absolute value $1$. If $f$ is a schlicht function and $|a\_n|=n$ for some

$n\backslash geq\; 2$, 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

:$f(z)=z+z^2\; =\; (z+1/2)^2\; -\; 1/4$

shows: it is holomorphic on the unit disc and satisfies $|a\_n|\backslash leq\; n$ for all $n$, but it is not injective since

$f(-1/2+z)\; =\; f(-1/2-z)$.

A survey of the history is given by [http://kobra.bibliothek.uni-kassel.de/bitstream/urn:nbn:de:hebis:34-200604038936/1/prep0513.pdf Koepf (2007)].

{{harvtxt|Bieberbach|1916}} proved $|a\_2|\backslash leq\; 2$, and stated the conjecture that $|a\_n|\backslash leq\; n$. {{harvtxt|Löwner|1917}} and {{harvtxt|Nevanlinna|1921}} independently proved the conjecture for starlike functions.

Then Charles Loewner ({{harvtxt|Löwner|1923}}) proved $|a\_3|\backslash leq\; 3$, using the Löwner equation. His work was used by most later attempts, and is also applied in the theory of Schramm–Loewner evolution.

{{harvtxt|Littlewood|1925|loc=theorem 20}} proved that $|a\_n|\backslash leq\; en$ for all $n$, showing that the Bieberbach conjecture is true up to a factor of $e=2.718\backslash ldots$ Several authors later reduced the constant in the inequality below $e$.

If $f(z)=z+\backslash cdots$ is a schlicht function then $\backslash varphi(z)\; =\; z(f(z^2)/z^2)^\{1/2\}$ is an odd schlicht function.

{{harvs|txt|authorlink=Raymond Paley|last=Paley|author2-link=John Edensor Littlewood|last2=Littlewood|year=1932}} showed that its Taylor coefficients satisfy $b\_k\backslash leq\; 14$ 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 {{harvtxt|Fekete|Szegő|1933}}, who showed there is an odd schlicht function with $b\_5=1/2+\backslash exp(-2/3)=1.013\backslash ldots$, and that this is the maximum possible value of $b\_5$. Isaak 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 $f$ is not a Koebe function (for which the $b\_\{2k+1\}$ are all $1$). 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 {{harvtxt|Robertson|1936}}.

The Robertson conjecture states that if

:$\backslash phi(z)\; =\; b\_1z+b\_3z^3+b\_5z^5+\backslash cdots$

is an odd schlicht function in the unit disk with $b\_1=1$ then for all positive integers $n$,

:$\backslash sum\_\{k=1\}^n|b\_\{2k+1\}|^2\backslash le\; n.$

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 {{harvtxt|Garabedian|Schiffer|1955}} proved $|a\_4|\backslash leq\; 4$, {{harvtxt|Ozawa|1969}} and {{harvtxt|Pederson|1968}} proved $|a\_6|\backslash leq\; 6$, and {{harvtxt|Pederson|Schiffer|1972}} proved $|a\_5|\backslash leq\; 5$.

{{harvtxt|Hayman|1955}} 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 schlicht function on the unit disk, and for all positive integers $n$,

:$\backslash sum^n\_\{k=1\}\; (n-k+1)(k|\backslash gamma\_k|^2-1/k)\backslash le\; 0$

where the logarithmic coefficients $\backslash gamma\_n$ of $f$ are given by

:$\backslash log(f(z)/z)=2\; \backslash sum^\backslash infty\_\{n=1\}\backslash gamma\_nz^n.$

{{harvtxt|Milin|1977}} 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 {{harvtxt|de Branges|1987}} proved $|a\_n|\backslash leq\; n$ for all $n$.

The proof uses a type of Hilbert space 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 {{harv|Milin|1977}} on logarithmic coefficients. This was already known to imply the Robertson conjecture {{harv|Robertson|1936}} about odd univalent functions, which in turn was known to imply the Bieberbach conjecture about schlicht functions {{harv|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 {{harvtxt|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.

De Branges proved the following result, which for $\backslash nu=0$ implies the Milin conjecture (and therefore the Bieberbach conjecture).

Suppose that $\backslash nu\; >\; -3/2$ and $\backslash sigma\_n$ are real numbers for positive integers $n$ with limit $0$ and such that

:$\backslash rho\_n=\backslash frac\{\backslash Gamma(2\backslash nu+n+1)\}\{\backslash Gamma(n+1)\}(\backslash sigma\_n-\backslash sigma\_\{n+1\})$

is non-negative, non-increasing, and has limit $0$. Then for all Riemann mapping functions $F(z)=z+\backslash cdots$ univalent in the unit disk with

:$\backslash frac\{F(z)^\backslash nu-z^\backslash nu\}\; \{\backslash nu\}=\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; a\_nz^\{\backslash nu+n\}$

the maximum value of

:$\backslash sum\_\{n=1\}^\backslash infty(\backslash nu+n)\backslash sigma\_n|a\_n|^2$

is achieved by the Koebe function $z/(1-z)^2$.

A simplified version of the proof was published in 1985 by Carl FitzGerald and Christian Pommerenke ({{harvtxt|FitzGerald|Pommerenke|1985}}), and an even shorter description by Jacob Korevaar ({{harvtxt|Korevaar|1986}}). A very short proof avoiding use of the inequalities of Askey and Gasper was later found by Lenard Weinstein ({{harvtxt|Weinstein|1991}}).

• Grunsky matrix
• Fekete–Szegő inequality
• Schwarz lemma

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