# Symmetric Polynomials and Symmetric Mean Inequalities

Keywords:
symmetric means, symmetric polynomials, arithmetic-geometric mean inequality

### Abstract

We prove generalized arithmetic-geometric mean inequalities for quasi-means arising from symmetric polynomials. The inequalities are satisfied by all positive, homogeneous symmetric polynomials, as well as a certain family of non-homogeneous polynomials; this family allows us to prove the following combinatorial result for marked square grids.

Suppose that the cells of a $n \times n$ checkerboard are each independently filled or empty, where the probability that a cell is filled depends only on its column. We prove that for any $0 \leq \ell \leq n$, the probability that each column has at most $\ell$ filled sites is less than or equal to the probability that each row has at most $\ell$ filled sites.