More symmetric polynomials related to p-norms

TL;DR

This paper introduces new symmetric polynomials related to p-norms, extending previous results and providing formulas for matrix eigenvalues, enhancing understanding of inequalities between symmetric functions and norms.

Contribution

The paper presents two new families of symmetric polynomials that generalize elementary symmetric polynomials and relate to p-norm inequalities, including formulas involving matrix eigenvalues.

Findings

New families of symmetric polynomials introduced.

Generalizations of p-norm inequalities established.

Formulas for polynomials in terms of matrix entries derived.

Abstract

It is known that the elementary symmetric polynomials $e_k(x)$ have the property that if $ x, y \in [0,\infty)^n$ and $e_k(x) \leq e_k(y)$ for all $k$, then $||x||_p \leq ||y||_p$ for all real $0\leq p \leq 1$, and moreover $||x||_p \geq ||y||_p$ for $1\leq p \leq 2$ provided $||x||_1 =||y||_1$. Previously the author proved this kind of property for $p>2$, for certain polynomials $F_{k,r}(x)$ which generalize the $e_k(x)$. In this paper we give two additional generalizations of this type, involving two other families of polynomials. When $x$ consists of the eigenvalues of a matrix $A$, we give a formula for the polynomials in terms of the entries of $A$, generalizing sums of principal $k \times k$ subdeterminants.