An operator inequality related to Jensen's inequality

For bounded non-negative operators A and B, Furuta showed 0 less than or equal toA less than or equal toB implies Ar/2 B-s Ar/2 less than or equal to (Ar/2 B-t Ar/2) s+r/t+r (0 less than or equal tor, 0 less than or equal tos less than or equal tot). We will extend this as follows: 0 less than or equal toA less than or equal toB(lambda)(!)C (0<<lambda><1) implies Ar/2 (<lambda>B-s + (1-lambda )C-s)Ar/2 less than or equal to {Ar/2 (lambdaB(t) + (1-lambda )C-t)Ar/2}s+r/t+r, where (BlambdaC)-C-! is a harmonic mean of B and C. The idea of the proof comes from Jensen's inequality for an operator convex function by Hansen-Pedersen.