This is an old problem from Ph.D Qualifying Exam of Complex Analysis.
Let $f$ be a holomorphic function in the open disc $D(0,2)$ of radius 2 centered at the origin and suppose that $|f(z)|=1$ whenever $|z|=1$, and $f(0)=0$. Prove that $|f'(z)|\ge 1$ if $|z|=1$.
My attempt: By maximum modulus principle, $|f(z)|< 1$ when $|z|<1$. Therefore, by Schwarz lemma, $|f(z)|\le |z|$ if $|z|\le 1$. Since $|f(z)|=1$ when $|z|=1$, I guess something similar to the Mean Value Theorem would hold, but I have no idea how to figure it out. Does anyone have ideas?
Thanks in advance!