I was reading the solution of a problem in the book "Berkeley problems in mathematics", and last part of the proof is unclear to me.
It somehow proves using Maximum principle that the rational function with given properties has constant magnitude on closed unit disk. Then, it immediately concludes that the function must be constant everywhere. I really don't get this. I'm asking this way in case there is a general fact about meromorphic functions, and this is just a special case. If that's not the case, I can give more details about what things the problem is specifically dealing with. I would appreciate any help. Thanks!