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Unformatted text preview: f ( z ) on C-[0 , 1]. State Liouvilles Theorem and use it to prove that f cannot be extended to [0 , 1] in such a way to make f an entire function. 4. (20 pts) a) (5 pts) Give a careful statement of the Schwarz Lemma. b) (15 pts) Prove that any analytic function f that maps the unit disc into itself, but is not one-to-one, must satisfy | f (0) | < 1. (Note, we do NOT assume that f (0) = 0 here.) 5. (20 pts) Suppose f is analytic on a neighborhood of the closed unit disc. If | f ( z ) | < 1 when | z | = 1, prove that there must exist at least one point z with | z | < 1 such that f ( z ) = z ....
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This document was uploaded on 01/25/2012.
- Spring '09