Koebe growth theorem
= Koebe growth theorem
{c}
{title2=$r/(1+r)^2\leq|f(z)|\leq r/(1-r)^2$}
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 $f(r\mathbb D)$.