A counterexample to the question proposed by Yanagi-Furuichi-Kuriyama on matrix inequalities and related counterexamples (Q1774980)

The author answers in the negative a question by \textit{K. Yanagi, S. Furuichi} and \textit{K. Kuriyama} [Linear Algebra Appl. 395, 351--359 (2005; Zbl 1067.94006)]. The question asks whether it is always true, for positive matrices \(A\leq I\) and \(B\leq I\), that \[ (A+B)^{1/2} \Bigl(A(\log\,A)^2+B(\log\,B)^2\Bigr) (A+B)^{1/2}\geq(A\,\log A+B\,\log B)^2 \] and simultaneously \[ \Bigl(A(\log\,A)^2+ B(\log\,B)^2\Bigr)^{1/2}(A+B)\Bigl(A(\log\,A)^2+ B(\log\,B)^2\Bigr)^{1/2} \geq(A\,\log A+B\,\log B)^2. \] The author shows two \(3\times3\) positive matrices \(A\leq I\) and \(B\leq I\) such that neither of both inequalities is satisfied. In the rest of the paper similar (slightly more general) problems are proposed, and counterexamples to all of them are shown, in a fashion similar to the first counterexample.