Problem 1602 (difficulty: 8/10)

Let \(\displaystyle f\) be a holomorphic function on the disc \(\displaystyle |z|<1+\varepsilon\). Prove that

\(\displaystyle \log |f(0)|\le \frac1{2\pi}\int_0^{2\pi}\log|f(e^{it})|\dt.\)

When does equality hold?

Give me another random problem!