# Bloch’s constant

Bloch’s theorem can be stated in the following way:

###### Bloch’s Theorem.

Let $\mathrm{F}$ be the set of all functions^{} $f$ holomorphic on a region containing the
closure of the disk $$ and satisfying
$f\mathit{}\mathrm{(}\mathrm{0}\mathrm{)}\mathrm{=}\mathrm{0}$ and ${f}^{\mathrm{\prime}}\mathit{}\mathrm{(}\mathrm{0}\mathrm{)}\mathrm{=}\mathrm{1}$. For each $f\mathrm{\in}\mathrm{F}$ let $\beta \mathit{}\mathrm{(}f\mathrm{)}$
be the supremum of all numbers $r$ such that there is a disk
$S\mathrm{\subset}D$ on which $f$ is injective and $f\mathit{}\mathrm{(}S\mathrm{)}$ contains a disk
of radius $r$. Let $B$ be the infimum of all $\beta \mathit{}\mathrm{(}f\mathrm{)}$, for $f\mathrm{\in}\mathrm{F}$. Then $B\mathrm{\ge}\mathrm{1}\mathrm{/}\mathrm{72}$.

The constant $B$ is usually referred to as Bloch’s constant. Nowadays, better bounds are known and, in fact, it has been conjectured that $B$ has the following tantalizing form

$$B=\frac{\mathrm{\Gamma}(1/3)\cdot \mathrm{\Gamma}(11/12)}{\left(\sqrt{1+\sqrt{3}}\right)\cdot \mathrm{\Gamma}(1/4)}$$ |

where $\mathrm{\Gamma}(x)$ is the gamma function^{}.

## References

- 1 John B. Conway, Functions of One Complex Variable I, Second Edition, 1978, Springer-Verlag, New York.

Title | Bloch’s constant |
---|---|

Canonical name | BlochsConstant |

Date of creation | 2013-03-22 15:58:04 |

Last modified on | 2013-03-22 15:58:04 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 5 |

Author | alozano (2414) |

Entry type | Definition |

Classification | msc 32H02 |

Related topic | LandausConstant |