REVERSE OF THE GRAND FURUTA INEQUALITY AND ITS APPLICATIONS

We shall give a norm inequality equivalent to the grand Furuta inequality, and moreover show its reverse as follows: Let A and B be positive operators such that 0 < m <= B <= M for some scalars 0 < m < M and h :(-) M/m > 1. Then parallel to A(1/2) {A(-t/2) (A(r/2) B((r-t){(p-t) s+r}/1-t+r) A(r/2))(1/s) A(-t/2)}(1/p) A(1/2)parallel to <= K(h(r-t), (p-t)s+r/1-t+r)(1/ps) parallel to A(1-t+r/2) Br-t A(1-t+r/2) parallel to((p-t)s+r/ps(1-t+r)) for 0 <= t <= 1, p >= 1, s >= 1 and r >= t >= 0, where K(h,p) is the generalized Kantorovich constant. As applications, we consider reverses related to the Ando-Hiai inequality.