More symmetric polynomials related to p-norms

Abstract

It is known that the elementary symmetric polynomials $e_k(x)$ have the property that if $x,y\in [0,\infty {)}^n$ and $e_k(x)\le e_k(y)$ for all $k$, then $||x|{|}_p\le ||y|{|}_p$ for all real $0\le p\le 1$, and moreover $||x|{|}_p\ge ||y|{|}_p$ for $1\le p\le 2$ 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.

arXiv comment27 pages