We show the following result: Let A be a positive operator satisfying 0 < m1(H) <= A <= M1(H) for some scalars m, M with m < M and Phi be a normalized positive linear map, then
Phi (A(-1)) <= Phi (m(A-M1H/M-m) Mm1H-A/M-m) <= (M + m)(2)/4Mm Phi(A)(-1).
Besides, we prove that the second inequality in the above can be squared.
