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Koebe 1/4 theorem (Theorem)
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
$\displaystyle f({\mathbb{D}}) \supseteq \{ w \mid \lvert w \rvert < 1/4 \} .$    

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.

Bibliography

1
John B. Conway. Functions of One Complex Variable II. Springer-Verlag, New York, New York, 1995.



"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
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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 4273 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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