Function order of positive operators based on the Mond-Pečarić method

We shall show function order preserving operator inequalities under general setting, based on Kantorovich type inequalities for convex functions due to Mond-Pev{; ; c}; ; ari'{; ; c}; ; : Let $A$ and $B$ be positive operators on a Hilbert space $H$ satisfying $MIgeq Bgeq mI>0$. Let $f(t)$ be a continuous convex function on $[m, M]$. If $g(t)$ is a continuous increasing convex function on $[m, M]cup {; ; mathsf{; ; Sp}; ; }; ; (A)$, then for a given $alpha >0$begin{; ; equation*}; ; Ageq Bgeq 0 quad text{; ; implies}; ; quad alpha g(A)+beta Igeq f(B) end{; ; equation*}; ; where $beta =max _{; ; mleq tleq M}; ; {; ; f(m)+(f(M)-f(m))(t-m)/(M-m)-alpha g(t) }; ; $. As applications, we shall extend Kantorovich type operator inequalities by Furuta, Yamazaki and Yanagida, and present operator inequalities on the usual order and the chaotic order via Ky Fan-Furuta constant. Among others, we show the following inequality: If $Ageq B>0$ and $MIgeq Bgeq mI>0$, then begin{; ; equation*}; ; frac{; ; M^{; ; p-1}; ; }; ; {; ; m^{; ; q-1}; ; }; ; A^qgeq frac{; ; (q-1)^{; ; q-1}; ; }; ; {; ; q^q}; ; frac{; ; (M^p-m^p)^q}; ; {; ; (M-m)(mM^p-Mm^p)^{; ; q-1}; ; }; ; A^qgeq B^p end{; ; equation*}; ; holds for all $p>1$ and $q>1$ such that $qm^{; ; p-1}; ; leq frac{; ; M^p-m^p}; ; {; ; M-m}; ; leq qM^{; ; p-1}; ; $.