The derivative of a holomorphic function on the boundary of the unit circle Let $f$ be holomorphic on $D(0,1)\cup \{1\}$, and $f(0)=0$, $f(1)=1$, $f(D(0,1))\subset D(0,1)$. Prove that $|f'(1)|\geq 1$.
I have no idea. Maybe $f'(\xi)=[f(1)-f(0)]/(1-0)$ for some $\xi \in (0,1)$, and the maximal modules principle applies? This is not exact.

Very strong hint:  You can actually prove that $|f'(1)| \geq \dfrac{2}{1+|f'(0)|}$, and the result then follows from Schwarz's lemma.  Establishing the above inequality will use some ideas from the proof of Schwarz lemma.  Seems very very tricky to me.  I would not have been able to solve when I was a first year graduate student... maybe there is a simpler way than what I found however.  Perhaps expanding in power series about 0 and 1 and solving systems of equations will give it to you.  In any case the Schwarz lemma must enter into the picture somehow.
