Problem 1784 (difficulty: 7/10)

Assume that \(\displaystyle f\) is regular on the unit disc and satisfies \(\displaystyle |f(z)|<1\). Show that

\(\displaystyle \frac{|f'(z)|}{1-|f(z)|^2} \le \frac1{1-|z|^2} . \)

Give me another random problem!