Abstract
In this paper, we employ the concept of operator means as well as some operator techniques to establish new operator Bellman and operator H\"{o}lder type inequalities. Among other results, it is shown that if $\mathbf{A}=(A_t)_{t\in \Omega}$ and $\mathbf{B}=(B_t)_{t\in \Omega}$ are continuous fields of positive invertible operators in a unital $C^*$-algebra ${\mathscr A}$ such that $\int_{\Omega}A_t\,d\mu(t)\leq I_{\mathscr A}$ and $\int_{\Omega}B_t\,d\mu(t)\leq I_{\mathscr A}$, and if $\omega_f$ is an arbitrary operator mean with the representing function $f$, then
\begin{align*}
\left(I_{\mathscr A}-\int_{\Omega}(A_t \omega_f B_t)\,d\mu(t)\right)^p
\geq\left(I_{\mathscr A}-\int_{\Omega}A_t\,d\mu(t)\right) \omega_{f^p}\left(I_{\mathscr A}-\int_{\Omega}B_t\,d\mu(t)\right)
\end{align*}
for all $0 < p \leq 1$, which is an extension of the operator Bellman inequality.
Keywords
Bellman inequality Cauchy-Schwarz inequality H\"{o}lder inequality operator mean Hadamard product continuous field of operators $C^*$-algebra.
References
- J. Aczél: Some general methods in the theory of functional equations in one variable, New applications of functional equations (Russian), Uspehi Mat. Nauk (N.S.) 11, 3 (69) (1956), 3–68.
- M. Bakherad: Some reversed and refined Callebaut inequalities via Kontorovich constant, Bull. Malays. Math. Sci. Soc., 41 (2) (2018), 765–777.
- M. Bakherad, A. Morassaei: Some operator Bellman type inequalities, Indag. Math. (N.S.), 26 (4) (2015), 646–659.
- M. Bakherad, A. Morassaei: Some extensions of the operator entropy type inequalities, Linear Multilinear Algebra, 67 (5) (2019), 871–885.
- E. F. Beckenbach, R. Bellman: Inequalities, Springer Verlag, Berlin (1971).
- R. Bellman: On an inequality concerning an indefinite form, Amer. Math. Monthly, 63 (1956), 101–109.
- J. L. Daz-Barrero, M. Grau-Sánchez, and P.G. Popescu: Refinements of Aczél, Popoviciu and Bellman’s inequalities, Comput. Math. Appl., 56 (2008), 2356–2359.
- S. Dragomir: Some additive reverses of Callebaut and Hölder inequalities for isotonic functionals, Constr. Math. Anal., 6 (4) (2023), 249–259.
- J. I. Fujii: The Marcus-Khan theorem for Hilbert space operators, Math. Japon., 41 (3) (1995), 531–535.
- M. Fujii, Y.O. Kim, and R. Nakamoto: A characterization of convex functions and its application to operator monotone functions, Banach J. Math. Anal., 8 (2) (2014), 118–123.
- T. Furuta, J. Mi´ci´c Hot, J. Peˇcari´c, and Y. Seo: Mond–Peˇcari´c method in operator inequalities, Element, Zagreb (2005).
- M. Gürdal, M. Alomari: Improvements of some Berezin radius inequalities, Constr. Math. Anal., 5 (3) (2022), 141–153.
- F. Hansen, I. Peri´c, and J. Peˇcari´c: Jensen’s operator inequality and its converses, Math. Scand., 100 (1) (2007), 61–73.
- E. Heinz: Beiträge zur Störungstheorie der Spektralzerlegung (German), Math. Ann., 123 (1951), 415–438.
- Z. Hu, A. Xu: Refinements of Aczél and Bellman’s inequalities, Comput. Math. Appl., 59 (9) (2010), 3078–3083.
- F. Kubo, T. Ando: Means of positive linear operators, Math. Ann., 246 (1980), 205–224.
- J. Mi´ci´c, J. Peˇcari´c, and J. Peri´c: Extension of the refined Jensen’s operator inequality with condition on spectra, Ann. Funct. Anal., 3 (1) (2011), 67–85.
- F. Mirzapour, M.S. Moslehian, and A. Morassaei: More on operator Bellman inequality, Quaest. Math., 37 (1) (2014), 9–17.
- M. Morassaei, F. Mirzapour, and M.S. Moslehian: Bellman inequality for Hilbert space operators, Linear Algebra Appl., 438 (2013), 3776–3780.
- M.S. Moslehian: Operator Aczél inequality, Linear Algebra Appl., 434 (8) (2011), 1981–1987.
- G.K. Pedersen: Analysis Now, Springer-Verlag, New York (1989).
- T. Popoviciu: On an inequality, Gaz. Mat. Fiz. Ser. A., 11 (64) (1959), 451–461.
- G. P. H. Styan: Hadamard product and multivariate statistical analysis, Linear Algebra Appl., 6 (1973), 217–240.
- S. Wada: On some refinement of the Cauchy–Schwarz inequality, Linear Algebra Appl., 420 (2-3) (2007), 433–440.
- S.Wu, L. Debnath. A new generalization of Aczèls inequality and its applications to an improvement of Bellmans inequality, Appl. Math. Lett., 21 (6) (2008), 588–593.
Abstract
References
- J. Aczél: Some general methods in the theory of functional equations in one variable, New applications of functional equations (Russian), Uspehi Mat. Nauk (N.S.) 11, 3 (69) (1956), 3–68.
- M. Bakherad: Some reversed and refined Callebaut inequalities via Kontorovich constant, Bull. Malays. Math. Sci. Soc., 41 (2) (2018), 765–777.
- M. Bakherad, A. Morassaei: Some operator Bellman type inequalities, Indag. Math. (N.S.), 26 (4) (2015), 646–659.
- M. Bakherad, A. Morassaei: Some extensions of the operator entropy type inequalities, Linear Multilinear Algebra, 67 (5) (2019), 871–885.
- E. F. Beckenbach, R. Bellman: Inequalities, Springer Verlag, Berlin (1971).
- R. Bellman: On an inequality concerning an indefinite form, Amer. Math. Monthly, 63 (1956), 101–109.
- J. L. Daz-Barrero, M. Grau-Sánchez, and P.G. Popescu: Refinements of Aczél, Popoviciu and Bellman’s inequalities, Comput. Math. Appl., 56 (2008), 2356–2359.
- S. Dragomir: Some additive reverses of Callebaut and Hölder inequalities for isotonic functionals, Constr. Math. Anal., 6 (4) (2023), 249–259.
- J. I. Fujii: The Marcus-Khan theorem for Hilbert space operators, Math. Japon., 41 (3) (1995), 531–535.
- M. Fujii, Y.O. Kim, and R. Nakamoto: A characterization of convex functions and its application to operator monotone functions, Banach J. Math. Anal., 8 (2) (2014), 118–123.
- T. Furuta, J. Mi´ci´c Hot, J. Peˇcari´c, and Y. Seo: Mond–Peˇcari´c method in operator inequalities, Element, Zagreb (2005).
- M. Gürdal, M. Alomari: Improvements of some Berezin radius inequalities, Constr. Math. Anal., 5 (3) (2022), 141–153.
- F. Hansen, I. Peri´c, and J. Peˇcari´c: Jensen’s operator inequality and its converses, Math. Scand., 100 (1) (2007), 61–73.
- E. Heinz: Beiträge zur Störungstheorie der Spektralzerlegung (German), Math. Ann., 123 (1951), 415–438.
- Z. Hu, A. Xu: Refinements of Aczél and Bellman’s inequalities, Comput. Math. Appl., 59 (9) (2010), 3078–3083.
- F. Kubo, T. Ando: Means of positive linear operators, Math. Ann., 246 (1980), 205–224.
- J. Mi´ci´c, J. Peˇcari´c, and J. Peri´c: Extension of the refined Jensen’s operator inequality with condition on spectra, Ann. Funct. Anal., 3 (1) (2011), 67–85.
- F. Mirzapour, M.S. Moslehian, and A. Morassaei: More on operator Bellman inequality, Quaest. Math., 37 (1) (2014), 9–17.
- M. Morassaei, F. Mirzapour, and M.S. Moslehian: Bellman inequality for Hilbert space operators, Linear Algebra Appl., 438 (2013), 3776–3780.
- M.S. Moslehian: Operator Aczél inequality, Linear Algebra Appl., 434 (8) (2011), 1981–1987.
- G.K. Pedersen: Analysis Now, Springer-Verlag, New York (1989).
- T. Popoviciu: On an inequality, Gaz. Mat. Fiz. Ser. A., 11 (64) (1959), 451–461.
- G. P. H. Styan: Hadamard product and multivariate statistical analysis, Linear Algebra Appl., 6 (1973), 217–240.
- S. Wada: On some refinement of the Cauchy–Schwarz inequality, Linear Algebra Appl., 420 (2-3) (2007), 433–440.
- S.Wu, L. Debnath. A new generalization of Aczèls inequality and its applications to an improvement of Bellmans inequality, Appl. Math. Lett., 21 (6) (2008), 588–593.
Details
| Primary Language | English |
|---|---|
| Subjects | Operator Algebras and Functional Analysis |
| Journal Section | Articles |
| Authors | |
| Early Pub Date | March 6, 2024 |
| Publication Date | March 15, 2024 |
| Submission Date | February 12, 2024 |
| Acceptance Date | March 6, 2024 |
| Published in Issue | Year 2024 Volume: 7 Issue: 1 |
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