An operator inequality related to Jensen’s inequality

Abstract:

For bounded non-negative operators $A$ and $B$, Furuta showed \[ 0\leq A \leq B \ \textrm {implies } \ A^{\frac {r}{2}}B^sA^{\frac {r}{2}} \leq (A^{\frac {r}{2}}B^t A^{\frac {r}{2}})^{\frac {s+r}{t+r}} \ \ (0\leq r, \ 0\leq s \leq t).\] We will extend this as follows: $0\leq A\leq B \underset {\lambda }{!}C$ $(0<\lambda <1)$ implies \[ A^{\frac {r}{2}}(\lambda B^s+ (1-\lambda )C^s)A^{\frac {r}{2}} \leq \{A^{\frac {r}{2}} (\lambda B^t+ (1- \lambda )C^t) A^{\frac {r}{2}}\}^{\frac {s+r}{t+r}} ,\] where $B \underset {\lambda }{!}C$ 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.