A new Newton-type inequality and the concavity of a class of k-trace functions

In this paper, we prove a new Newton-type inequality that generalizes Newton's inequality. With this new inequality, we prove the concavity of a class of $k$-trace functions, $A\mapsto \ln \mathrm{Tr}_k[\exp(H+\ln A)]$, on the convex cone of all positive definite matrices. $\mathrm{Tr}_k[A]$ denotes the $k_{\mathrm{th}}$ elementary symmetric polynomial of the eigenvalues of $A$. As an application, we use the concavity of these $k$-trace functions to derive expectation estimates on the sum of the $k$ largest (or smallest) eigenvalues of sum of random matrices.