Solution
= Solution
The <Koebe growth theorem> states that every <normalized univalent function> $\phi\in\mathcal S$ satisfies
$$
\boxed{\frac{|z|}{(1+|z|)^2}\leq|\phi(z)|\leq\frac{|z|}{(1-|z|)^2}\qquad(z\in\mathbb D)}.
$$
Both bounds are sharp. Rotations of the <Koebe function>, $\phi_\theta(z)=z/(1-e^{i\theta}z)^2$, attain them along suitable radial directions.