On some operator inequalities

A real-valued continuous function f(2) defined on a (finite or infinite) interval of the real line is said to be operator monotone if, for any pair of bounded selfadjoint operators A, B on an infinite dimensional Hilbert space, A > B implies f (A) > f(B). Here f(A) and f(B) are defined according to the usual functional calculus, and > refers to the order relation induced by the cone of positive (semi-definite) operators. Operator monotony is quite a restrictive condition. In fact, for p > 0 the function f(2): = 2 p is operator monotone on the half-line [0, c~) if and only if 0 1 the function f(2) = 2 p possesses some property akin to operator monotony. In this direction Furuta I-3] recently observed that if r > 0, p > 0, q > 1 satisfies (1 + 2r)q > p + 2r then A > B > 0 implies A tp § 2,)/q > (ArBPA r) l/~. If p = 2r and q = 2, the requirement is automatically satisfied, hence A p > (Ap/2BPAP/2)I/2 whenever A > B > 0. When A is invertible, this means that