# Journal of Operator Theory

Volume 77, Issue 1, Winter 2017  pp. 77-86.

On the Ando-Hiai-Okubo trace inequality

Authors:  Mostafa Hayajneh (1), Saja Hayajneh (2), and Fuad Kittaneh (3)
Author institution:(1) Department of Mathematics, Yarmouk University, Irbid, Jordan
(2) Department of Mathematics, The University of Jordan, Amman, Jordan
(3) Department of Mathematics, The University of Jordan, Amman, Jordan

Summary: Let $A$ and $B$ be positive semidefinite matrices. It is shown that $|\mathrm{Tr}(A^{w}B^{z}A^{1-w}B^{1-z})|\leqslant \mathrm{Tr}(AB)$ for all complex numbers $w,z$ for which $|\mathrm{Re}\, w$ $-\frac{1}{2}|+|\mathrm{Re}\, z-\frac{1}{2} |\leqslant \frac{1}{2}.$ This is a generalization of a trace inequality due to T.~Ando, F. Hiai, and K. Okubo for the special case when $w,z$ are real numbers, and a recent trace inequality proved by T. Bottazzi, R. Elencwajg, G. Larotonda, and A. Varela when $w=z$ with $\frac{1}{4}\leqslant \re z\leqslant \frac{3}{4}$. As a consequence of our new trace inequality, we prove that $\|A^{w}B^{z}+B^{1-\overline{z}}A^{1-\overline{w}}\|_{2}\leqslant \|A^{w}B^{z}+A^{1-\overline{w}}B^{1-\overline{z}}\|_{2}$ for all complex numbers $w,z$ for which $|\re w-\frac{1}{2}|+|\re z-\frac{1}{2} |\leqslant \frac{1}{2}.$ This is a generalization of a recent norm inequality proved by M. Hayajneh, S. Hayajneh, and F. Kittaneh when $w,z$ are real numbers.

DOI: http://dx.doi.org/10.7900/jot.2015dec23.2096
Keywords: unitarily invariant norm, Hilbert--Schmidt norm, Schatten $p$-norm, trace, positive semidefinite matrix, inequality