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On a matrix trace inequality due to Ando, Hiai and Okubo. (English) Zbl 1368.15019

Summary: T. Ando et al. [Math. Inequal. Appl. 3, No. 3, 307–318 (2000; Zbl 0959.15015)] have proved that inequality \(\mathfrak{Re}\, \mathrm{tr}\,A^{p_1}B^{q_1}\cdots A^{p_k}B^{q_k}\leq \mathrm{tr}\,A^{p_1+\dots+p_k}B^{q_1+\dots+q_k}\) is valid for all positive semidefinite matrices \(A\), \(B\) and those nonnegative real numbers \(p_1, q_1, \dots, p_k, q_k\) which satisfy certain additional conditions. We give an example to show that this inequality is not valid for all collections of \(p_1, q_1, \dots, p_k\), \(q_k \geq 0\). We also study related trace inequalities.

MSC:

15A45 Miscellaneous inequalities involving matrices
15A15 Determinants, permanents, traces, other special matrix functions

Citations:

Zbl 0959.15015
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References:

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