Bloch'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 $D$

where $B > \dfrac 1 {72}$ is an absolute constant.

The lower bound $\dfrac 1 {72}$ is not the best possible.

Source of Name

This entry was named for André Bloch.