On a matrix trace inequality due to Ando, Hiai and Okubo

Abstract

Ando et al. have proved that inequality \(\Re \mathfrak{e}trA^{p_1 } B^{q_1 \ldots } A^{p_k } B^{q_k } \leqslant trA^{p_1 + \ldots + p_k } B^{q_1 + \ldots + q_k }p\) is valid for all positive semidefinite matrices A,B and those nonnegative real numbers p1, q1,..., pk, qk which satisfy certain additional conditions. We give an example to show that this inequality is not valid for all collections of p1, q1,..., pk, qk ≥ 0. We also study related trace inequalities.