A general form preserving the operator order for convex functions

Abstract

We show a general form preserving the operator order for convex functions based on the Mond-Pecaric method as follows: Let A and B be positive operators on a Hilbert space H satisfying M1H B m1H > 0 . Let f (t) be a continuous convex function on [m, M] . If g(t) is a continuous increasing convex function on [m, M] ∪ Sp(A) , then for a given α > 0 A B 0 implies αg(A) + βIH f (B) where β = maxm t M{f (m) + [(f (M) − f (m))/(M − m)](t − m) − αg(t)} . We extend Kantorovich type operator inequalities via the Ky Fan-Furuta constant as applications. Among others, we show the following inequality: If A B > 0 and M1H B m1H > 0 , then Mp−1 mq−1 A q (q − 1) q−1 qq (Mp − mp)q (M − m)(mMp − Mmp)q−1 A q Bp holds for all p > 1 and q > 1 . Mathematics subject classification (2000): 47A30, 47A63.