New log-majorization results concerning eigenvalues and singular values and a complement of a norm inequality

Abstract

The purpose of this paper is to establish new log-majorization results concerning eigenvalues and singular values which generalize some previous work related to a conjecture and an open question which were presented by R. Lemos and G. Soares in [11]. In addition, we present a complement of a unitarily invariant norm inequality which was conjectured by R. Bhatia, Y. Lim and T. Yamazaki in [4], and recently proved by T.H. Dinh, R. Dumitru and J.A. Franco in [5] for the Schatten p-norm with 1 ≤ p ≤ ∞.