Inequalities of Furuta and Mond-Pečarić

Furuta gives extensions of inequalities due to Ky Fan and Mond-Pečarić which are associated with Holder-McCarthy and Kantorovich type inequalities. In this paper, inspired by Furuta's idea, we shall generalize a theorem by Mond-Pečarić on the converse of Jensen's inequality. It explains us Furuta's generalizations well. As applications, we shall show general difference and ratio inequalities that can be given for several positive operators on a Hilbert space and positive definite Hermitian matrices.