I'm working on this problem
"Let $f$ be an entire function. Suppose $f$ restricted to the unit disk is a bijection. Prove that $f$ is a rotation."
My attempt: It is tempting to use Schwarz lemma. Let $T$ be the linear transformation that maps the unit disk to itself and $T(f(0))=0$ (If I remember it right, $T=\frac{z-f(0)}{1-\overline{f(0)}z}$). Then $|Tf(z)|\leq 1$ in the unit disk and $T(f(0))=0$.
So, if I can show that $|Tf(z)|=|z|$ for some nonzero $z$, we are done by the lemma. I want to say $Tf$ achieves maximum on the unit circle. So if the maximum is 1, $|T(f(z))|=1=|z|$ for some $z$ there. However I am stuck here.