Holomorphic functions with constant modulus
Problem Statement: Suppose that is holomorphic on a domain with constant. Prove that is a constant map.
Ideas: If then . Suppose , . So where with is a holomorphic branch of ; if such a branch did not exist then would not be holomorphic on . I tried applying the definition of the complex derivative directly to show that to no avail. If it were shown that for any in that it would be done, but I don't see how to make that work either.
Any hints, ideas, or a solution are welcome at this point. Thank you.