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)| r. Show...

$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|*, |=| > r. Show that f is a polynomial of degree at most k. (b) Let f and g be analytic in |z| < 1. Suppose also that they have no zeros there, and that |S(2)| = \g(2)| on |z| = 1. Show that there is a constant a with la| = 1 such that f(2) = ag(2). $

Extracted text: 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|*,="" |="|"> r. Show that f is a polynomial of degree at most k. (b) Let f and g be analytic in |z| < 1.="" suppose="" also="" that="" they="" have="" no="" zeros="" there,="" and="" that="" |s(2)|="\g(2)|" on="" |z|="1." show="" that="" there="" is="" a="" constant="" a="" with="" la|="1" such="" that="" f(2)="">

Jun 03, 2022

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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)| r. Show...

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