On Zero-Sector Reducing Operators

TL;DR

This paper introduces the first example of a zero-sector reducing operator, proving a Jensen-disc type theorem for polynomials with zeros in a sector and constructing linear operators that shrink the zero sector.

Contribution

It establishes a Jensen-disc type theorem for sector-zero polynomials and constructs the first known zero-sector reducing linear operators.

Findings

Proved a Jensen-disc type theorem for polynomials with zeros in a sector.

Constructed linear operators that reduce the zero sector of polynomials.

First example of 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 $T\colon\mathbb{R}[z]\to\mathbb{R}[z]$ which map polynomials with their zeros in a closed convex sector $|\arg z| \leq \theta<\pi/2$ to polynomials with zeros in a smaller sector $|\arg z| \leq \gamma<\theta$. We, therefore, provide the first example of a zero-sector reducing operator.