Some functions reversing the order of positive operators
Let \(A\) and \(B\) be bounded linear operators on a complex Hilbert space. Under the condition \(A\geq B>0\), the authors prove the following: (i) if \(f\) is a convex function, then \(\alpha g(B)+\beta I\geq f(A)\) holds for some \(\alpha\), \(\beta\) and \(g\), (ii) if \(f\) is a concave function, then \( f(B)\geq \alpha g(A)+\beta I\) holds for some \(\alpha\), \(\beta\) and \(g\), and (iii) if \(f\) is a convex or concave function, then there exist positive numbers \(\alpha_{1}\) and \(\alpha_{2}\) such that \[ \alpha_{1}f\left(\sum_{j=1}^{k}\omega_{j}A_{j}\right)\leq \sum_{j=1}^{k}\omega_{j}f(A_{j})\leq \alpha_{2}f\left(\sum_{j=1}^{k}\omega_{j}A_{j}\right), \] where \(\omega_{j}\) are positive numbers satisfying \( \sum_{j=1}^{k}\omega_{j}=1\) and \(A_{j}\) are positive invertible operators. (i) and (ii) are reverse inequalities of those shown in [Linear Algebra Appl. 360, 15--34 (2004; Zbl 1030.47013)].
scientific article; zbMATH DE number 2136677