Many classical inequalities such as those in title can be extended to familiar combinatorial symmetric function families such as the elementary, homogeneous, power sum, and Schur polynomials. In fact all known instances of these extended inequalities hold in a strong form ("Y-positivity") which appears to be new. This talk will report on details of this project, including results and open problems. The Y-version of Newton's Inequalities is proved using an application of the FKG inequality to a poset of 2-rowed standard Young tableaux. This is joint work with Mark Skandera and Jonathan Lima.