Some functions reversing the order of positive operators

As a complement to our previous results about the function preserving the operator order, we shall show the following reversing version: 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 decreasing convex function on [m,M]∪Sp(A), then for a given α > 0A⩾B⩾0impliesαg(B)+βI⩾f(A),where β = maxm⩽t⩽M{f(m) + (f(M) − f(m))(t − m)/(M − m) − αg(t)}. Our main result is to classify complementary inequalities on power means of positive operators. As a matter of fact, we determine real constants α1 and α1 such that α2Mk[s](A;ω)⩽Mk[r](A;ω)⩽α1Mk[s](A;ω) if r ⩽ s, where Mk[r](A;ω):=(∑j=1kωjAjr)1/r (r∈R⧹{0}) is weighted power mean of positive operators Aj, Sp(Aj)⊆[m,M] for some scalars 0 < m < M and ωj∈R+ such that ∑j=1kωj=1 (j = 1, … ,k).