New Newton-type inequality and a unified perspective on Maclaurin's inequality via the inclusion relations of cones

The classical Newton-Maclaurin's inequalities for the elementary symmetric means are well known. In this paper, we derive a new class of Newton-type inequalities and generalize Maclaurin's inequality by connecting the elementary symmetric functions with their counterparts. Our idea is mainly motivated by the inclusion relations of G$\mathring{\text{a}}$rding's cones and $k$-convex cones.