Landau's Theorem

Theorem
Let $f: \C \to \C$ be a holomorphic function in the unit disk $\cmod z \le 1$.

Let $\cmod {\map {f'} 0} = 1$.

Then there exists:
 * a disk $D$ of radius $B$
 * an analytic function $\phi$ in $D$ such that $\map f {\map \phi z} = z$ for all $z$ in $D$

such that $B$ is an absolute constant where:
 * $B < \pi \sqrt 2^{1/4} \dfrac {\map \Gamma {1/3} } {\map \Gamma {1/4} } \paren {\dfrac {\map \Gamma {11/12} } {\map \Gamma {1/12} } }^{1/2}$.

Also see

 * Bloch's Theorem
 * Value of Landau's Constant
 * Value of Bloch's Constant