Bloch’s constant

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

Bloch’s Theorem.

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

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

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

where $\Gamma(x)$ 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 BlochsConstant 2013-03-22 15:58:04 2013-03-22 15:58:04 alozano (2414) alozano (2414) 5 alozano (2414) Definition msc 32H02 LandausConstant