I am working on a problem from an old complex analysis qual, and have run across the following problem:
Let $f(z)$ be holomorphic on $D(0,1)$, such that $|f(z)|\leq 1$ for all $|z|<1$. If $f(0) = f'(0) = 0$, prove $|f''(0)|\leq 2$.
After working through it for awhile, I realized we definitely needs Schwarz' Lemma for the proof. I'm just unsure of how to handle the second derivative. I think working with the cases of $f(z) = 1$ and $f(z) < 1$ is the way to start.