Complementary inequalities to inequalities of Jensen and Ando based on the Mond-Pečarić method (Q1590648)

The authors treat inequalities of the form \(\Phi(f(A)) \geq \alpha f(\Phi(A)) + \beta I\), where \(A\) is a positive linear operator on a Hilbert space \(H\), \(I\) is the identity on \(H\), \(\Phi\) is a unital positive linear map operating on the bounded linear operators of \(H\), \(f\) is a nonnegative concave function on a compact interval of the reals, and \(\alpha\), \(\beta\) are certain constants. (In fact, \(A\) is subject to some extra conditions which are omitted here.) This result includes and extends several particular cases that have been established individually before. Also, when \(f\) is defined appropriately, the result can be transferred as to obtain inequalities on operator-means and Hadamard products. Cf., among others, \textit{T. Ando} [Linear Algebra Appl. 26, 203-241 (1979; Zbl 0495.15018)], \textit{J. Micic, Y. Seo, S.-E. Takahasi} and \textit{M. Tominaga} [Math. Inequal. Appl. 2, No. 1, 83-111 (1999; Zbl 0924.47013)], and \textit{J. I. Fujii} [Math. Jap. 41, No. 3, 531-535 (1995; Zbl 0826.47016)].