Ando–Hiai inequality and Furuta inequality

Let A, B > 0, the positive operators on a Hilbert space. Ando–Hiai inequality (AH) and Furuta inequality (F) are given as follows:equation(AH)A,B>0;A♯αB⩽I⇒Ar♯αBr⩽Iforr⩾1,andequation(F)A⩾B>0⇒A-r♯1+rp+rBp⩽B⩽A.We show these inequalities imply each other.