Generalization of the Kantorovich type operator inequalities via grand Furuta inequality

In this note we show the characterization of the $delta$-order by means of a generalized Kantorovich constant via Grand Furuta inequality, which is an extension result of that from M.Fujii, E.Kamei, Y.Seo, {; ; it Kantorovich type operator inequalities via grand Furuta inequality}; ; , Sci. Math., {; ; bf 3}; ; (2000), 263--272. Among other, we show the following characterization of the $delta$-order: Let $A, B$ be positive invertible operators on a Hilbert space $H$ satisfying $M I geq A geq m I >0$ and $N I geq B geq n I>0$. Then the following statements are mutually equivalent for each $delta in [0, 1]$: {; ; rm (i)}; ; quad $A^{; ; delta}; ; geq B^{; ; delta}; ; $. {; ; rm (ii)}; ; quad $K(n^r, N^r, 1+frac{; ; p- delta}; ; {; ; r}; ; , 1+frac{; ; q- delta}; ; {; ; r}; ; )A^q geq B^p$ for all $p > delta$, $q > delta$ and $r > delta$. {; ; rm (iii)}; ; quad $overline{; ; K}; ; (m^r, M^r, 1+frac{; ; q- delta}; ; {; ; r}; ; , 1+frac{; ; p- delta}; ; {; ; r}; ; )A^q geq B^p$ for all $p > delta$, $q > delta$ and $r > delta$ where the case $delta=0$ means the chaotic order $log A geq log B$.