(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 S(2)| = |g(2)| on |z| = 1. Show that there is a constant a with la| = 1 such that f(2) = ag(2).
(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 S(2)| = |g(2)| on |z| = 1. Show that there is a constant a with la| = 1 such that f(2) = ag(2).
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.3: Factorization In F [x]
Problem 17E: Suppose that f(x),g(x), and h(x) are polynomials over the field F, each of which has positive...
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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
\f(=)| < C]z|*,
|=| 2 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
|f(2)| = \g(2)| on |z| = 1. Show that there is a constant a with |a| = 1 such that f(z) = ag(z).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F942d8395-ce39-44af-9e4d-fbdcfc13f43e%2F6f65b9c2-3dde-46d4-b115-d0c827bbf915%2F5l6weti_processed.jpeg&w=3840&q=75)
Transcribed Image 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
\f(=)| < C]z|*,
|=| 2 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
|f(2)| = \g(2)| on |z| = 1. Show that there is a constant a with |a| = 1 such that f(z) = ag(z).
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