Koebe 1/4 theorem

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The Koebe 1/4 theorem states that the image of an injective analytic function $f:\mathbb D\to\mathbb C$ from the unit disk $\mathbb D$ onto a subset of the complex plane contains the disk whose center is $f (0)$ and whose radius is $|f\,'(0)|/4$ . The theorem is named after Paul Koebe, who conjectured the result in 1907. The theorem was proven by Ludwig Bieberbach in 1914. The Koebe function $f (z) = z / (1 - z) 2$ shows that the constant $1 / 4$ in the theorem cannot be improved.