Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2015 • Question 8

Question

Math 6210 • Real Analysis Spring 2015 • Question 8

Let \(f\) be an entire function. Suppose that there exist \(A>0\), \(R>0\), and a positive integer \(k\) such that \[ |f(z)|\le A|z|^k\ln |z| \] for \(|z|\ge R\). Show that \(f\) is a polynomial of degree at most \(k\).

Raw LaTeX

Let \(f\) be an entire function. Suppose that there exist \(A>0\),
\(R>0\), and a positive integer \(k\) such that
\[
|f(z)|\le A|z|^k\ln |z|
\]
for \(|z|\ge R\). Show that \(f\) is a polynomial of degree at most \(k\).

Topic Tags

Entire Functions and Liouville-Type Theorems