PlanetMath: Landau's constant

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Landau's constant

(Definition)

We suggest that the reader reads first the entry on Bloch's constant. Let $\mathcal{F}$ be the set of all functions holomorphic on a region containing the closure of the disk $D=\{z\in\mathbb{C}:\vert z\vert<1\}$ and satisfying and . For each $f\in\mathcal{F}$ let $\lambda(f)$ be the supremum of all numbers such that there is a disk $S\subset D$ such that contains a disk of radius (notice that here we don't require to be injective on ).

Definition 1 Landau's constant is defined by

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

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

Bibliography

1: John B. Conway, Functions of One Complex Variable I, Second Edition, 1978, Springer-Verlag, New York.

"Landau's constant" is owned by alozano.

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See Also: Bloch's constant

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Cross-references: strictly, injective, radius, contains, supremum, closure, region, holomorphic, functions, Bloch's constant

This is version 1 of Landau's constant, born on 2006-06-09.
Object id is 7984, canonical name is LandausConstant.
Accessed 582 times total.

Classification:

AMS MSC:

32H02 (Several complex variables and analytic spaces :: Holomorphic mappings and correspondences :: Holomorphic mappings, embeddings and related questions)

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