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

Autori
Mićić, Jadranka ; Pečarić, Josip ; Seo, Yuki

Izvornik
Linear algebra and its applications (0024-3795) 360 (2003), 1; 15-34

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
function order; positive operators; Mond-Pečarić method

Sažetak
We shall show function order preserving operator inequalities under general setting, based on Kantorovich type inequalities for convex functions due to Mond-Pe\v{; ; c}; ; ari\'{; ; c}; ; : Let $A$ and $B$ be positive operators on a Hilbert space $H$ satisfying $MI\geq B\geq mI>0$. 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*}; ; A\geq B\geq 0 \quad \text{; ; implies}; ; \quad \alpha g(A)+\beta I\geq f(B) \end{; ; equation*}; ; where $\beta =\max _{; ; m\leq t\leq 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 $A\geq B>0$ and $MI\geq B\geq mI>0$, then \begin{; ; equation*}; ; \frac{; ; M^{; ; p-1}; ; }; ; {; ; m^{; ; q-1}; ; }; ; A^q\geq \frac{; ; (q-1)^{; ; q-1}; ; }; ; {; ; q^q}; ; \frac{; ; (M^p-m^p)^q}; ; {; ; (M-m)(mM^p-Mm^p)^{; ; q-1}; ; }; ; A^q\geq 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}; ; $.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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