Landau Constant

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

Let $F$ be the set of Complex analytic functions $f$ defined on an open region containing the closure of the unit disk $D=\{z: \vert z\vert<1\}$ satisfying $f(0)=0$ and $df/dz(0)=1$. For each $f$ in $F$, let $l(f)$ be the Supremum of all numbers $r$ such that $f(D)$ contains a disk of radius $r$. Then

$$L\equiv{\rm inf}\{l(f): f\in F\}.$$

This constant is called the Landau constant, or the Bloch-Landau Constant. Robinson (1938, unpublished) and Rademacher (1943) derived the bounds

$${\textstyle{1\over 2}}<L\leq{\Gamma({\textstyle{1\over 3}})\... ...over 6}})\over\Gamma({\textstyle{1\over 6}})}=0.5432588\ldots,$$

where $\Gamma(z)$ is the Gamma Function, and conjectured that the second inequality is actually an equality,

$$L={\Gamma({\textstyle{1\over 3}})\Gamma({\textstyle{5\over 6}})\over\Gamma({\textstyle{1\over 6}})}=0.5432588\ldots.$$

See also Bloch Constant

References

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/bloch/bloch.html

Rademacher, H. ``On the Bloch-Landau Constant.'' Amer. J. Math. 65, 387-390, 1943.