 Koebe growth theorem

For a normalized univalent function $f$ on the unit disc, put $r=|z|<1$. The displayed sharp two-sided bounds hold. Rotations $z/(1-e^{i\theta }z{)}^2$ of the Koebe function attain the bounds along suitable radial directions. In particular, bounding the modulus controls the radius of a disk containing each smaller image $f(r\mathbb{D})$.