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Bloch's constant
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(Definition)
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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 $D=\{z\in\mathbb{C}:|z|<1\}$ 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 $$B=\frac{\Gamma(1/3)\cdot \Gamma(11/12)}{\left(\sqrt{1+\sqrt{3}}\right)\cdot \Gamma(1/4)}$$ where $\Gamma(x)$ is the gamma function.
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- John B. Conway, Functions of One Complex Variable I, Second Edition, 1978, Springer-Verlag, New York.
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"Bloch's constant" is owned by alozano.
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Cross-references: gamma function, bounds, infimum, radius, contains, injective, numbers, supremum, closure, region, holomorphic, functions, Bloch's theorem
There is 1 reference to this entry.
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 2221 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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Pending Errata and Addenda
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