(a) Let f be analytic in a bounded region D and its boundary C, such that |f(z)| = 1
on C. Show that f has at least one zero inside D, unless f is a constant.
(b) Let f(z) be an analytic function in a region D except for one simple pole and
assume |f(z)| = 1 on the boundary of D. Prove that every value a with |a| > 1 is
taken by f(z) inside D once and once only.
Please see the attached file.
Here we use a basic fact that if is analytic in a region , then is a constant if and only if is a constant. The "if" part can be verified by C-R equations.
Now in this problem, suppose is not a constant and has no zeros in the region . Let . Since is analytic, then is analytic. So both and can not obtain its maximum value ...
This is two proofs regarding analytic functions.