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    Analytic Function Proofs on Bounded Regions

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    (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.

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    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.