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.

Proof

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Also see

• Definition:Bloch's Constant

• Lower Bound of Bloch's Constant
• Upper Bound of Bloch's Constant

• Landau's Theorem

Source of Name

This entry was named for André Bloch.

Sources

• 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,433012701 \ldots < B < 0,4719 \ldots$