Characterizing non-constant entire functions with modulus $1$ on the unit circle 
Is there a characterization of the nonconstant entire functions $f$ that satisfy $|f(z)|=1$ for all $|z|=1$?

Clearly, $f(z)=z^n$ works for all $n$. Also, it's not difficult to show that if $f$ is such an entire function, then $f$ must vanish somewhere inside the unit disk. What else can be said about those functions?
Thank you.

Partial answer. 
If $|f(z)|=1$ for all $|z|=1$ and $f$ is entire function, then $f:\mathbb D \to \mathbb D$, by the Maximum Modulus Theorem. 
Using Schwarz-Lemma we can characterize, at least, the conformal ones.
( See Functions of one Complex Variable, Conway 2ed pag. 131 )
