8. (Liouville's theorem, Cauchy estimates and the maximum modulus principle) (a) Suppose that f is entire, and that for some positive integer k and some constants C,r > 0 it holds that \S(2)|< C\z|*, |=| 2r. Show that f is a polynomial of degree at most k. (b) Let f and g be analytic in |2| < 1. Suppose also that they have no zeros there, and that |f(2)| = |g(2)| on |2| = 1. Show that there is a constant a with |a| = 1 such that f(2) = ag(2).

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