Inequalities of Furuta and Mond-Pečarić on the Hadamard product (Q1580810)

In a previous paper [Math. Inequal. Appl. 2, No. 1, 83-111 (1999; Zbl 0924.47013)] the authors discussed inequalities of the form \[ (f(A)x, x)\leq\alpha g((Ax, x))+ \beta, \] where \(\alpha\) is a given real number and \(\beta\) is appropriately chosen. Here \(x\) is any unit vector and \(f\), \(g\) are real continuous functions, the first one convex. The present paper deals with similar inequalities over the Hadamard product. For instance, if \(A\), \(B\) are two strictly positive operators and \(f\) is a real, convex, supermultiplicative function one has for certain \(g\): \[ f(A)* f(B)\leq\alpha g(A* B)+\beta I. \] The inequality changes direction when \(f\) is concave.