I'm having difficulties with a question from Complex Analysis (Gamelin). The question has been asked before, but I still have some difficulties with it. It asks to show that a function continuous on unit disk and its boundary and analytic on the unit open disk is a finite Blaschke product if its modulus on the boundary of the disk is one. The solutions I have read suggested that one divide the function by the finite Blaschke product with zeros identical to zeros of f. I first of all do not understand why f should have a finite number of zeros. Secondly, would it not follow by the maximum and minimum modulus principle that f has constant modulus on the entire disk since its maximum and minimum moduli on the boundary of the disk are one? Then, how can f acquire any zeros whatsoever? Am I misusing the max/min mod principle here?
As well, how does dividing the function by the finite Blaschke product with zeros identical to the zeros of the function help? The claim was that the function is now by the max mod principle constant (??). I still don't see how this follows.