Lower Bound of Bloch's Constant
Theorem
Bloch's constant has a lower bound as follows:
- $\dfrac {\sqrt 3} 4 + \dfrac 2 {10 \, 000} \le B$
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
- 1996: Huaihui Chen and Paul M. Gauthier: On Bloch's constant (Journal d'Analyse Mathématique Vol. 69: pp. 275 – 291)