The best possibility of the grand Furuta inequality

Let A, B is an element of B(H) be invertible bounded linear operators on a Hilbert space H satisfying O less than or equal to B less than or equal to A, and let p, r, s, t be real numbers satisfying 1 < s, 0 < t < 1, t less than or equal to 1, 1 less than or equal to p. Furuta showed that if 0 < alpha less than or equal to 1 - t + r/(p - t)s + r, then {A(r/2)(A(-t/2) B-p A(-t/2))(s) A(r/2)}(alpha) less than or equal to A({(p-t)s+r}alpha). This inequality is called the grand Furuta inequality, which interpolates the Furuta inequality (t = 0) and the Ando-Hiai inequality (t = 1, r = s). In this paper, we show the grand Furuta inequality is best possible in the following sense: that is, if 1 - t + r/(p - t)s + r < alpha, then there exist invertible matrices A, B with O less than or equal to B less than or equal to A which do not satisfy {A(r/2) (A(-t/2) B-p A(-t/2))(s) A(r/2)}(alpha) less than or equal to A({(p-t)s+r}alpha).