Multiplier sequences and logarithmic mesh

TL;DR

This paper proves that finite multiplier sequences do not decrease the logarithmic mesh of polynomials with all real roots of the same sign, revealing a new property of these operators in preserving root spacing.

Contribution

It establishes a novel property of multiplier sequences, showing they preserve the logarithmic mesh of polynomials with real roots, which was previously unknown.

Findings

Multiplier sequences do not decrease the logarithmic mesh.

The logarithmic mesh is defined as the minimal quotient of consecutive roots.

This property holds for polynomials with all roots real and of the same sign.

Abstract

In this note we prove a new result about (finite) multiplier sequences, i.e. linear operators acting diagonally in the standard monomial basis of R[x] and sending polynomials with all real roots to polynomials with all real roots. Namely, we show that any such operator does not decrease the logarithmic mesh when acting on an arbitrary polynomial having all roots real and of the same sign. The logarithmic mesh of such a polynomial is defined as the minimal quotient of its consecutive roots taken in the non-decreasing order.