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

08/10/2018

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by   De Huang, et al.

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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, ATr_k[(H+ A)], on the convex cone of all positive definite matrices. Tr_k[A] denotes the k_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.