It sounds as if they are asking you to rediscover the theory of Blaschke products.
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).
Here's a bare outline of how the proof goes. The set G of all complex rational functions which have modulus 1 on the unit circle is a group (under pointwise multiplication). For each complex number a with |a|<1, the function is in G.
Now suppose that f is in G. If f has no zeros in the unit ball then f must be constant. (That is the tricky part of the argument. According to the Wikipedia page that I linked to in my previous comment above, "this fact is a consequence of the maximum principle for harmonic functions, applied to the harmonic function log(|f(z)|)".)
If f does have zeros on the unit ball, call them a_1, a_2, ..., a_n (counted according to their multiplicity). Then is a function in G with no zeros in the unit ball and is therefore constant. Hence for some number with .
That's the structure theorem for functions in G. Whether or not it's fair to ask it as an exam question depends on the students' background, I guess. I would say it's a pretty tough assignment for any student who hasn't previously seen the result.