Editing De Branges's theorem (section)

==Schlicht functions==

The normalizations

:<math>a_0=0 \ \text{and}\ a_1=1</math>

mean that

:<math> f(0)=0\ \text{and}\ f'(0)=1 . </math>

This can always be obtained by an [[affine transformation]]: starting with an arbitrary injective holomorphic function <math>g</math> defined on the open unit disk and setting

:<math>f(z)=\frac{g(z)-g(0)}{g'(0)}.</math>

Such functions <math>g</math> are of  interest because they appear in the  [[Riemann mapping theorem]].

A '''schlicht function''' is defined as an analytic function <math>f</math> that is one-to-one and satisfies <math>f(0)=0</math> and <math>f'(0)=1</math>.  A  family of schlicht functions are the [[rotated Koebe function]]s

:<math>f_\alpha(z)=\frac{z}{(1-\alpha z)^2}=\sum_{n=1}^\infty n\alpha^{n-1} z^n</math>

with <math>\alpha</math> a [[complex number]] of [[absolute value]] <math>1</math>. If <math>f</math> is a schlicht function and <math>|a_n|=n</math> for some 
<math>n\geq 2</math>, then <math>f</math> is a rotated Koebe function.

The condition of de Branges' theorem is not sufficient to show the function is schlicht, as the function
:<math>f(z)=z+z^2 = (z+1/2)^2 - 1/4</math>
shows: it is holomorphic on the unit disc and satisfies <math>|a_n|\leq n</math> for all <math>n</math>, but it is not injective since
<math>f(-1/2+z) = f(-1/2-z)</math>.