Characterizations of $delta$–order associated with Kantorovich type operator inequalities

In this note, we obtain more precise estimations than the constants are given in the paper by M.Fujii, E.Kamei and Y.Seo, {; ; it Kantorovich type operator inequalities via grand Furuta inequality}; ; , Sci. Math., {; ; bf 3}; ; (2000), 263--272. Among other, we show that the following statements are mutually equivalent for each $delta in (0, 1]$: (i) $ K(m^{; ; frac{; ; (p-delta)s}; ; {; ; n}; ; }; ; , M^{; ; frac{; ; (p-delta)s}; ; {; ; n}; ; }; ; , n+1)^{; ; frac{; ; 1}; ; {; ; s}; ; }; ; A^{; ; p}; ; geq B^{; ; p}; ; $ for any $n>0$, $s geq 1$, $p geq delta$ with $ (p-delta )s geq n delta $. (ii) $K(m^{; ; delta}; ; , M^{; ; delta}; ; , frac{; ; p}; ; {; ; delta}; ; ) A^{; ; p}; ; geq B^{; ; p}; ; qquad mbox{; ; for any $p geq delta$ }; ; $. For each $delta in (0, 1]$ $$K(m^{; ; frac{; ; (p-delta)s}; ; {; ; n}; ; }; ; , M^{; ; frac{; ; (p-delta)s}; ; {; ; n}; ; }; ; , n+1)^{; ; frac{; ; 1}; ; {; ; s}; ; }; ; geq K(m^{; ; delta}; ; , M^{; ; delta}; ; , frac{; ; p}; ; {; ; delta}; ; )$$ holds for any $n>0$, $s geq 1$, $p geq delta$ such that $ (p-delta )s geq n delta $.