Editing De Branges's theorem (section)

==De Branges's proof==
The proof uses a type of [[Hilbert space]] of [[entire function]]s. The study of these spaces grew into a sub-field of complex analysis and the spaces have come to be called [[de Branges space]]s. 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, <!--the Rogosinski conjecture {{harv|Rogosinski|1943}} about subordinate 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 polynomial]]s, 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 ([[St. Petersburg Department of Steklov Institute of Mathematics of Russian Academy of Sciences|Leningrad Department of Steklov Mathematical Institute]]) when de Branges visited in 1984.

De Branges proved the following result, which for <math>\nu=0</math> implies the Milin conjecture (and therefore the Bieberbach conjecture). 
Suppose that <math>\nu > -3/2</math> and <math>\sigma_n</math> are real numbers for positive integers <math>n</math> with limit <math>0</math> and such that
:<math> \rho_n=\frac{\Gamma(2\nu+n+1)}{\Gamma(n+1)}(\sigma_n-\sigma_{n+1}) </math>
is non-negative, non-increasing, and has limit <math>0</math>. Then for all Riemann mapping functions <math>F(z)=z+\cdots</math> univalent in the unit disk with
:<math>\frac{F(z)^\nu-z^\nu} {\nu}= \sum_{n=1}^{\infty} a_nz^{\nu+n}</math>
the maximum value of 
:<math>\sum_{n=1}^\infty(\nu+n)\sigma_n|a_n|^2</math>
is achieved by the Koebe function <math>z/(1-z)^2</math>.

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}}).