Extend algebraic theory to inhomogeneous symmetric-polynomial inequalities

Develop an extension of the existing algebraic theory of inequalities for symmetric polynomials that can prove the inhomogeneous inequality relating the fixed-point measure of a uniformly random permutation to the normalized second elementary symmetric mean of the square roots of non-negative variables.

Background

The paper proves an inhomogeneous inequality for elementary symmetric polynomials: for non-negative real numbers a_1, ..., a_n, the expected product of the variables indexed by the fixed points of a uniformly random permutation is at least the normalized second elementary symmetric mean of their square roots. The authors emphasize that existing algebraic methods for symmetric-polynomial inequalities are primarily designed for homogeneous inequalities.

The unresolved issue is to determine how the existing algebraic theory can be generalized so that it applies to this new class of inhomogeneous inequalities and, in particular, yields a proof of the paper's main result. The authors suggest that such an extension could lead to further discoveries, but do not provide one.