Koebe quarter theorem
Statement in complex analysis / From Wikipedia, the free encyclopedia
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In complex analysis, a branch of mathematics, the Koebe 1/4 theorem states the following:
Koebe Quarter Theorem. The image of an injective analytic function from the unit disk onto a subset of the complex plane contains the disk whose center is and whose radius is .
The theorem is named after Paul Koebe, who conjectured the result in 1907. The theorem was proven by Ludwig Bieberbach in 1916. The example of the Koebe function shows that the constant in the theorem cannot be improved (increased).
A related result is the Schwarz lemma, and a notion related to both is conformal radius.