An operator monotone function \(\frac{t\log t-t+1}{\log^2 t}\) and strictly chaotic order (Q5957185)

In this paper, firstly, the author obtains two simplified proofs of operator monotonicity of NEWLINE\[NEWLINE f(t)=\frac{t\log t-t+1}{\log^{2}t}NEWLINE\]NEWLINE and its dual function NEWLINE\[NEWLINE f^{*}(t)=\frac{t\log^{2}t}{t\log t-t+1}NEWLINE\]NEWLINE by using the Löwner-Heinz inequality.NEWLINENEWLINENEWLINESecondly, the author shows that for positive invertible operators \(A\) and \(B\) satisfying \(1\not\in \sigma(A)\cup\sigma(B)\), (i) \(\log A>\log B\) implies that there exists \(\beta\in ]0,1]\) such that \(f(A^{\alpha})>f(B^{\alpha})\) holds for all \(\alpha\in ]0,\beta[\), (ii) \( A> B\) implies that there exists \(\beta\in ]0,1]\) such that \(f(A^{\alpha})>f(B^{\alpha})\) holds for all \(\alpha\in ]0,\beta[\), and (iii) \( A\geq B\) implies that \(f(A^{\alpha})\geq f(B^{\alpha})\) holds for all \(\alpha\in ]0,1]\).NEWLINENEWLINENEWLINELastly, in contrast to the above results, the author points out that there exist positive invertible operators \(A\) and \(B\) such that \(\log A\geq \log B\) and \(f(A^{\alpha})\not\geq f(B^{\alpha})\) for any \(\alpha>0\).