Lipschitz type continuity for matrix means

Abstract

In this article, we study continuity type inequalities for unitarily invariant norms of several families of matrix means, to include Kubo-Ando, Rényi, and power means. In particular, if σ represents a mean of matrices, we consider inequalities of the form ‖AσC−BσC‖≤K‖f(|A−B|)‖, for some finite K. Moreover, we consider inequalities of the form, ‖AσC−BσC‖≤Kf(‖A−B‖), where f is operator monotone and f(0)=0.