Suppose $f(z)$ is entire function such that $|f(z)|=1$ on $C(0,1)$. Show that $f(z)=k z^n$ for some constant $k$,$|k|=1$, and for some nonnegative integer $n$.
I was thinking to apply Blaschke product by breaking the number of zeros of $f(z)$ inside the circle and outside the circle. I also have to consider the cases $f(z)$ might have infinitely many zero in the whole complex plane or it might never vanish. I am getting confuse how to start.
Any help would be appreciated!