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.

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