A theorem of Pólya on polynomials

Abstract

Among the many contributions of George Pólya to analysis, the following has always been Erdős’ favorite, both for the surprising result and for the beauty of its proof. Suppose that

$$ f(z) = z^{n} + b_{n-1}z^{n-1} + ... + b_{0} $$

is a complex polynomial of degree n ≥ 1 with leading coefficient 1. Associate with f(z) the set

$$ \mathcal{C} := \{z \in \mathbb{C}: |f(z)| \leq 2 \} $$

, that is, $\mathcal{C}$ is the set of points which are mapped under f into the circle of radius 2 around the origin in the complex plane. So for n = 1 the domain $\mathcal{C}$ is just a circular disk of diameter 4.