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Let f be an entire function such that |f(2)| < 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.

Let f be an entire function such that |f(2)| < 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.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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.
Transcribed Image Text: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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