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[parent] Bloch's constant (Definition)

Bloch's theorem can be stated in the following way:

Bloch's Theorem 1   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}:\vert z\vert<1\}$ and satisfying $ f(0)=0$ and $ f'(0)=1$. For each $ f\in\mathcal{F}$ let $ \beta(f)$ be the supremum of all numbers $ r$ such that there is a disk $ S\subset D$ on which $ f$ is injective and $ f(S)$ contains a disk of radius $ r$. Let $ B$ be the infimum of all $ \beta(f)$, for $ f\in \mathcal{F}$. Then $ B\geq 1/72$.

The constant $ B$ is usually referred to as Bloch's constant. Nowadays, better bounds are known and, in fact, it has been conjectured that $ B$ has the following tantalizing form

$\displaystyle B=\frac{\Gamma(1/3)\cdot \Gamma(11/12)}{\left(\sqrt{1+\sqrt{3}}\right)\cdot \Gamma(1/4)}$
where $ \Gamma(x)$ is the gamma function.

Bibliography

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



"Bloch's constant" is owned by alozano.
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See Also: Landau's constant


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Cross-references: gamma function, bounds, infimum, radius, contains, injective, numbers, supremum, closure, region, holomorphic, functions, Bloch's theorem
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This is version 2 of Bloch's constant, born on 2006-06-09, modified 2006-10-02.
Object id is 7983, canonical name is BlochsConstant.
Accessed 1508 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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