Some ways of constructing Furuta-type inequalities

Let σhσh be an operator mean associated with an operator monotone function h   and let A,BA,B be positive operators. We investigate the following Furuta-type inequality: For some fixed continuous function h,A≥B>0⇒A−rσφrXh−r≤B(r≥1), where XhXh is the positive solution of h(X)=B−1h(X)=B−1. The map (h,r)↦φr(h,r)↦φr plays a central role in constructing a Furuta-type inequality, similar to the role of the map (p,r)↦r+1p+r in the following part of the Furuta inequality: A≥B>0,p,r≥1⇒(Br2ApBr2)1+rp+r≥(Bp+r)1+rp+r. We obtain this map (h,r)↦φr(h,r)↦φr from the following Ando–Hiai type inequality:A,B>0,AσhB≤1⇒ArσhBr≤1(r≥1).