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

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,..., p_k, q_k which satisfy certain additional conditions. We give an example to show that this inequality is not valid for all collections of p₁, q₁,..., p_k, q_k ≥ 0. We also study related trace inequalities.