A quadratic Schwarz-type estimate from a derivative bound

Let \(f\) be analytic on the unit disk \(D\), with \[ f(0)=f'(0)=0, \qquad |f'(z)|<1\quad(z\in D). \] Show that \[ |f(z)|\le\frac{|z|^2}{2} \qquad(z\in D). \]

Hint: represent \(f(z)\) as a path integral along a convenient path.