If $f$ is entire and $|f|=1$ on $|z|=1$,then $f(z)=cz^n$ for some $c$.
First consider $g(z)=f(z)/\prod(z-a_i)/(1-\overline{a_i}z)$,where $a_i$ are zeros of $f(z)$.
Then I want to apply the maximum and minimum modulus theorem to argue that all $a_i$'s are zero. But what am I supposed to do? Do I need to first show that $g(z)$ is constant?