Positive Semidefinite Matrices, Exponential Convexity for Majorization, and Related Cauchy Means

Abstract

We prove positive semidefiniteness of matrices generated by differences deduced from majorization-type results which implies exponential convexity and log-convexity of these differences and also obtain Lyapunov's and Dresher's inequalities for these differences. We introduce new Cauchy means and show that these means are monotone.