On Ando’s inequalities for convex and concave functions

For positive semidefinite matrices A and B, Ando and Zhan proved the inequalities |||f(A) + f(B)||| ≥ |||f(A + B)||| and |||g(A) + g(B)||| ≤ |||g(A + B)|||, for any unitarily invariant norm, and for any non-negative operator monotone f on [0,∞) with inverse function g. These inequalities have very recently been generalised to non-negative concave functions f and non-negative convex functions g, by Bourin and Uchiyama, and Kosem, respectively. In this paper we consider the related question whether the inequalities |||f(A)− f(B)||| ≤ |||f(|A−B|)|||, and |||g(A)− g(B)||| ≥ |||g(|A−B|)|||, obtained by Ando, for operator monotone f with inverse g, also have a similar generalisation to nonnegative concave f and convex g. We answer exactly this question, in the negative for general matrices, and affirmatively in the special case when A ≥ ||B||. In the course of this work, we introduce the novel notion of Y -dominated majorisation between the spectra of two Hermitian matrices, where Y is itself a Hermitian matrix, and prove a certain property of this relation that allows to strengthen the results of Bourin-Uchiyama and Kosem, mentioned above.