We shall show function order preserving operator inequalities under general setting, based on Kantorovich type inequalities for convex functions due to Mond–Pečarić: 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 [m,M]∪Sp(A), then for a given α>0A⩾B⩾0impliesαg(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, thenMp−1mq−1Aq⩾(q−1)q−1qq(Mp−mp)q(M−m)(mMp−Mmp)q−1Aq⩾Bpholds for all p>1 and q>1 such thatqmp−1⩽Mp−mpM−m⩽qMp−1.