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.

I have got it using Schwarz lemma. It follows that $|f(z)|\leq |z|$. And hence 
$$f'(1)=\lim_{R\ni r\to 1^-}\frac{f(r)-f(1)}{r-1}$$ that $|f'(1)|\geq 1$. It is not easy as this. Since $f(r)$ may be complex-valued.
