# Upper Bound of Bloch's Constant

## Theorem

Bloch's constant has an upper bound as follows:

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

## Proof

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## 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 $\pi \sqrt 2^{1/4} \dfrac {\map \Gamma {1/3} } {\map \Gamma {1/4} } \paren {\dfrac {\map \Gamma {11/12} } {\map \Gamma {1/12} } }^{1/2}$.

## Sources

- 1937: Lars Valerian Ahlfors and Helmut Grunsky:
*Über die Blochsche Konstante*(*Math. Z.***Vol. 42**: pp. 671 – 673)