Koebe quarter theorem

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 ${\displaystyle f:\mathbf {D} \to \mathbb {C} }$ from the unit disk ${\displaystyle \mathbf {D} }$ onto a subset of the complex plane contains the disk whose center is ${\displaystyle f(0)}$ and whose radius is ${\displaystyle |f'(0)|/4}$.

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 ${\displaystyle 1/4}$ in the theorem cannot be improved (increased).

A related result is the Schwarz lemma, and a notion related to both is conformal radius.