Problem 1785 (difficulty: 6/10)

Let the roots of the regular function \(\displaystyle f: S(0, 1) \to S(0, 1)\) be \(\displaystyle a_1, \dots , a_n\). Show that

\(\displaystyle |f(z)|\le \prod\limits_{i=1}^n \left|\frac{a_i -z}{1-\overline{a_i}z}\right| \qquad (|z|<1).\)

Give me another random problem!