Function order of positive operators based on the Mond–Pe ari method

We shall show function order preserving operator inequalities under general setting, based on Kantorovich type inequalities for convex functions due to Mond–Pe ari : Let A and B be positive operators on a Hilbert space H satisfying MI B mI>0. Let f(t) be a continuous convex function on [m,M]. If g(t) is a continuous increasing convex function on , then for a given α>0 A B 0 implies αg(A)+βI f(B),where β=maxm t M{f(m)+(f(M)−f(m))(t−m)/(M−m)−αg(t)}. As applications, we shall extend Kantorovich type operator inequalities by Furuta, Yamazaki and Yanagida, and present operator inequalities on the usual order and the chaotic order via Ky Fan–Furuta constant. Among others, we show the following inequality: If A B>0 and MI B mI>0, then holds for all p>1 and q>1 such that