Let $f : {\bf C} \to {\bf C}$ be an entire function.
Assume that $|f(z)| = 1$ when $|z| = 1$.
Is it true that we must have $f = z \mapsto C z^m$ where $m$ is an integer and $|C| = 1$ ?*
I think the answer is yes. By maximum modulus, $f(z) \leq 1$ whenever $|z| \leq 1$. Likely Cauchy's theorem can be applied here perhaps after applying a suitable Mobius transform.
