Koebe Quarter Theorem
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Theorem
Let $f: \C \to \C$ be a schlicht function, that is, a univalent complex function such that $\map f 0 = 0$ and $\map {f'} 0 = 1$.
Then the image of the unit disk contains the closed disk of radius $\dfrac 1 4$.
Hence for any $w \in f \sqbrk {\Bbb D}$ we have that $\cmod w \le \dfrac 1 4$.
The constant $\dfrac 1 4$ is sharp and so cannot be improved.
Proof
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Sources
- 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,25$