Landau’s constant

We suggest that the reader reads first the entry on Bloch’s constant. Let $ℱ$ be the set of all functions $f$ holomorphic on a region containing theclosure of the disk and satisfying$f(0)=0$ and $f^{\prime }(0)=1$. For each $f\in ℱ$ let $\lambda (f)$be the supremum of all numbers $r$ such that there is a disk$S\subset D$ such that $f(S)$ contains a disk of radius $r$ (notice that here we don’t require $f$ to be injective on $S$).

Definition.

Landau’s constant $L$ is defined by

$$L=inf\{\lambda (f):f\in ℱ\}.$$

Let $B$ be Bloch’s constant. Then, clearly, $L\ge B$. The exact value of $L$ (as that of $B$) is not known but it has been shown that $0.5\le L\le 0.56$. In particular, it is known that $L$ is strictly greater than $B$.