# Analytic Function Proofs on Bounded Regions

(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.

https://brainmass.com/math/discrete-math/analytic-function-proofs-bounded-regions-107065

#### Solution Preview

Please see the attached file.

(a) Proof:

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 ...

#### Solution Summary

This is two proofs regarding analytic functions.