Quadratic derivative-bound estimate and equality case

Suppose \(f\) is analytic in the unit disk, \(f(0)=f'(0)=0\), and \[ |f'(z)|\le1 \] for every \(|z|<1\). Prove \[ |f(z)|\le \frac{|z|^2}{2}. \] Determine the functions for which equality can hold at some nonzero \(z\).