# Definition:Bloch's Constant

## Definition

Recall Bloch's 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 of $B$ is known as **Bloch's constant**.

## Also see

- Lower Bound of Bloch's Constant, where it is shown that $\dfrac {\sqrt 3} 4 + 2 \times 10^{-4} \le B$.

- Upper Bound of Bloch's Constant, where it is shown that $B \le \sqrt {\dfrac {\sqrt 3 -1} 2} \times \dfrac {\map \Gamma {\frac 1 3} \map \Gamma {\frac {11} {12} } } {\map \Gamma {\frac 1 4} }$

## Source of Name

This entry was named for André Bloch.

## Historical Note

The precise value of **Bloch's constant** is unknown.

André Bloch stated a lower bound for it of $\dfrac 1 {72}$.

However, it is known that $\dfrac 1 {72}$ is not the best possible value for it.

In their $1983$ work *Les Nombres Remarquables*, François Le Lionnais and Jean Brette give $\dfrac {\sqrt 3} 4$:

- $\dfrac {\sqrt 3} 4 \approx 0 \cdotp 43301 \, 2701 \ldots$

This sequence is A120011 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

The best value known at present is $\dfrac {\sqrt 3} 4 + \dfrac 2 {10 \, 000}$ which evaluates to approximately $0 \cdotp 43321 \, 2701$.

This was demonstrated by Huaihui Chen and Paul M. Gauthier in $1996$.

Lars Valerian Ahlfors and Helmut Grunsky demonstrated that:

- $B \le \sqrt {\dfrac {\sqrt 3 -1} 2} \times \dfrac {\map \Gamma {\frac 1 3} \map \Gamma {\frac {11} {12} } } {\map \Gamma {\frac 1 4} }$

and conjectured that this value is in fact the true value of $B$.

The number is given by François Le Lionnais and Jean Brette as $\pi \sqrt 2^{1/4} \dfrac {\map \Gamma {1/3} } {\map \Gamma {1/4} } \paren {\dfrac {\map \Gamma {11/12} } {\map \Gamma {1/12} } }^{1/2}$.