As I said I was taking an exam more than a month ago on Complex Analyis. One of the optional problems was (if I remember correcty):

"Find all complex rational functions which have modulos 1 on the unit circle"*.

Note: I cannot help you with this problem it uses a theorem which I never learned.

*)I was told that this problem was a favorite problem to give as an examination problem to graduate students in a certain event (which I forgot the name of).