# Landau’s constant

We suggest that the reader reads first the entry on Bloch’s constant. 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 $\lambda(f)$ 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

 $L=\inf\{\lambda(f):f\in\mathcal{F}\}.$

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

## References

• 1 John B. Conway, Functions of One Complex Variable I, Second Edition, 1978, Springer-Verlag, New York.
Title Landau’s constant LandausConstant 2013-03-22 15:58:07 2013-03-22 15:58:07 alozano (2414) alozano (2414) 4 alozano (2414) Definition msc 32H02 BlochsConstant