Some functions reversing the order of positive operators Original Research Article

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 greater-or-equal, slanted B greater-or-equal, slanted 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]union or logical sumSp(A), then for a given α > 0imagewhere β = maxmless-than-or-equals, slanttless-than-or-equals, slantM 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 image if r less-than-or-equals, slant s, where image (rset membership, variantR-45 degree rule 0 ) is weighted power mean of positive operators Aj, Sp(Aj)subset of or equal to[m,M] for some scalars 0 < m < M and ωjset membership, variantR+ such that image (j = 1, … ,k).