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 theclosure 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 diskof 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 beenconjectured 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.