8. Let f be an entire function such that f(z)| < A + B|z|" for all z E C, where A and B are positive real constants and n is a fixed non-negative integer. Show that f is a polynomial of degree at most n.

8. Let f be an entire function such that f(z)| < A + B|z|" for all z E C, where A and B are positive real constants and n is a fixed non-negative integer. Show that f is a polynomial of degree at most n.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.1: Polynomials Over A Ring

Problem 12E: a. Find a nonconstant polynomial in Z4[ x ], if one exists, that is a unit. b. Find a nonconstant...

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Question

8. Let f be an entire function such that |f(z)| < A + B\z|" for all z € C, where A and B are
positive real constants and n is a fixed non-negative integer. Show that f is a polynomial of
degree at most n.
9. Let P(z)
= z" + an-12n-1 + ..+ a1z + ao be a polynomial of degree n > 1 with a; # 0 for at
least one i. Prove that there exists a point z* such that 2*| = 1 and |P(z*)| > 1.

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