Landau's constantLandausConstant2006-06-09 12:57:522006-06-09 12:57:52DefinitionBloch's theorem% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Cl}{\operatorname{Cl}}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 $f'(0)=1$. 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 $S$).
\begin{defn}
Landau's constant $L$ is defined by
$$L=\inf \{ \lambda(f) : f\in \mathcal{F}\}.$$
\end{defn}
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 $B$.
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\bibitem{conway} John B. Conway, {\em Functions of One Complex
Variable I}, Second Edition, 1978, Springer-Verlag, New York.
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