On Zero-Sector Reducing Operators

Abstract

We prove a Jensen-disc type theorem for polynomials $p\in \mathbb{R}[z]$ having all their zeros in a sector of the complex plane. This result is then used to prove the existence of a collection of linear operators $T:\mathbb{R}[z]\to \mathbb{R}[z]$ which map polynomials with their zeros in a closed convex sector $|\arg z|\le \theta <\pi /2$ to polynomials with zeros in a smaller sector $|\arg z|\le \gamma <\theta$. We, therefore, provide the first example of a zero-sector reducing operator.

11 pages, 3 figures