Koebe Quarter Theorem

Theorem
Let $f: \C \to \C$ be a schlicht function, that is, a univalent complex function such that $\map f 0 = 0$ and $\map {f'} 0 = 1$.

Then the image of the unit disk contains the closed disk of radius $\dfrac 1 4$.

Hence for any $w \in f \sqbrk {\Bbb D}⁢$ we have that $\cmod w \le \dfrac 1 4$.

The constant $\dfrac 1 4$ is sharp and so cannot be improved.