On a trace inequality (Q1381267)

Generalizing trace inequalities of \textit{H. Kosaki} [Proc. Cent. Math. Appl. Aust. Natl. Univ. 29, 129-134 (1991; Zbl 0801.47012)] and \textit{T. Furuta} [Linear Algebra Appl. 235, 153-161 (1996; Zbl 0846.47014)], the author proves the following result. Let \(A,B\) be positive linear operators on a complex Hilbert space and let \(p,\alpha,s\in \mathbb{R}_+\), \(\beta\in \mathbb{R}\). Assume either (1) \(A\geq B\) and \(\beta\geq \max\{-\frac 12 (p+2\alpha)\), \(-\frac 12 (1+2\alpha)\}\), or (2) \(A,B\) are invertible with \(\log A\geq\log B\) and \(\beta\geq -\alpha\). Then, for any continuous increasing function \(f\) on \(\mathbb{R}_+\) with \(f(0)=0\), the trace inequality \(\text{Tr } f(A^\beta (A^\alpha B^p A^\alpha)^s A^\beta)\leq \text{Tr } f(A^{(p+ 2\alpha)s+ 2\beta)})\) holds, \(A\) being assumed invertible if \(\beta<0\).