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Koebe 1/4 theorem
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(Theorem)
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Theorem 1 (Koebe) Suppose $f$ is a schlicht function (univalent function on the unit disc such that $f(0) = 0$ and $f'(0) = 1$ ) and ${\mathbb{D}} \subset {\mathbb{C}}$ is the unit disc in the complex plane, then \begin{equation*} f({\mathbb{D}}) \supseteq \{ w \mid \lvert w \rvert < 1/4 \} . \end{equation*}
That is, if a univalent function on the unit disc maps 0 to 0 and has derivative 1 at 0, then the image of the unit disc contains the ball of radius $1/4$ . So for any $w \notin f({\mathbb{D}})$ we have that $\lvert w \rvert \geq 1/4$ . Furthermore, if we look at the Koebe function, we can see that the constant $1/4$ is sharp and cannot be improved.
- 1
- John B. Conway. Functions of One Complex Variable II. Springer-Verlag, New York, New York, 1995.
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"Koebe 1/4 theorem" is owned by jirka. [ full author list (2) ]
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See Also: schlicht functions
| Other names: |
Köbe 1/4 theorem, Koebe one-fourth theorem, Köbe one-fourth theorem |
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Cross-references: Koebe function, radius, ball, contains, image, derivative, maps, complex plane, unit disc, univalent function, schlicht function
There is 1 reference to this entry.
This is version 5 of Koebe 1/4 theorem, born on 2004-06-07, modified 2005-03-07.
Object id is 5896, canonical name is Koebe14Theorem.
Accessed 5760 times total.
Classification:
| AMS MSC: | 30C45 (Functions of a complex variable :: Geometric function theory :: Special classes of univalent and multivalent functions ) |
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Pending Errata and Addenda
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