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 $e_k(x) \leq e_k(y)$ 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.