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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 1996 • Question 6

Question

Math 6210 • Real Analysis Fall 1996 • Question 6

(a) Let \(f\) be an entire function. Assume that there exists a polynomial \(g\) such that for each \(z\in \mathbb C\), \(|f(z)|\le |g(z)|\). Show that \(f\) is also a polynomial.

(b) Can we draw the same conclusion if the function \(g\) is rational? In other words, let \(f\) be an entire function and \(g\) a rational function such that for each \(z\in \mathbb C\), which is not a pole of \(g\), \(|f(z)|\le |g(z)|\). Does it follow that \(f\) is a polynomial?

Raw LaTeX

(a) Let \(f\) be an entire function. Assume that there exists a
polynomial \(g\) such that for each \(z\in\mathbb C\), \(|f(z)|\le |g(z)|\). Show
that \(f\) is also a polynomial.

(b) Can we draw the same conclusion if the function \(g\) is rational? In other
words, let \(f\) be an entire function and \(g\) a rational function such that for
each \(z\in\mathbb C\), which is not a pole of \(g\), \(|f(z)|\le |g(z)|\). Does it
follow that \(f\) is a polynomial?

Topic Tags

Entire Functions and Liouville-Type Theorems Isolated Singularities and Residues Meromorphic Functions and Rational Functions