Editing De Branges's theorem (section)

==History ==
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 <math>|a_2|\leq 2</math>, and stated the conjecture that <math>|a_n|\leq n</math>. {{harvtxt|Löwner|1917}} and  {{harvtxt|Nevanlinna|1921}} independently proved the conjecture for [[Nevanlinna's criterion#Application to Bieberbach conjecture|starlike functions]].
Then [[Charles Loewner]] ({{harvtxt|Löwner|1923}}) proved <math>|a_3|\leq 3</math>, 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 <math>|a_n|\leq en</math> for all <math>n</math>, showing that the Bieberbach conjecture is true up to a factor of <math>e=2.718\ldots</math> Several authors later reduced the constant in the inequality below <math>e</math>.
 
If <math>f(z)=z+\cdots</math> is a schlicht function then <math>\varphi(z) = z(f(z^2)/z^2)^{1/2}</math> 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 <math> b_k\leq 14</math> for all <math>k</math>. They conjectured that <math>14</math> can be replaced by <math>1</math> 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 <math>b_5=1/2+\exp(-2/3)=1.013\ldots</math>, and that this is the maximum possible value of <math>b_5</math>. [[Isaak Milin]] later showed that <math>14</math> can be replaced by <math>1.14</math>, and Hayman showed that the numbers <math>b_k</math> have a limit less than <math>1</math> if <math>f</math> is not a Koebe function (for which the <math>b_{2k+1}</math> are all <math>1</math>). So the limit is always less than or equal to <math>1</math>, 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

:<math>\phi(z) = b_1z+b_3z^3+b_5z^5+\cdots</math>

is an odd schlicht function in the unit disk with <math>b_1=1</math> then for all positive integers <math>n</math>, 
:<math>\sum_{k=1}^n|b_{2k+1}|^2\le n.</math>

Robertson observed that his conjecture is still strong enough to imply the Bieberbach conjecture, and proved it for <math>n=3</math>. 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 <math>n</math>, in particular {{harvtxt|Garabedian|Schiffer|1955}} proved <math>|a_4|\leq 4</math>, {{harvtxt|Ozawa|1969}} and {{harvtxt|Pederson|1968}} proved <math>|a_6|\leq 6</math>, and {{harvtxt|Pederson|Schiffer|1972}} proved <math>|a_5|\leq 5</math>.

{{harvtxt|Hayman|1955}} proved that the limit of <math>a_n/n</math> exists, and has absolute value less than <math>1</math> unless <math>f</math> is a Koebe function. In particular this showed that for any <math>f</math> 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 <math>n</math>, 
:<math>\sum^n_{k=1} (n-k+1)(k|\gamma_k|^2-1/k)\le 0</math>

where the '''logarithmic coefficients''' <math>\gamma_n</math> of <math>f</math> are given by

:<math>\log(f(z)/z)=2 \sum^\infty_{n=1}\gamma_nz^n.</math>
  
{{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 <math>|a_n|\leq n</math> for all <math>n</math>.