Research Article
BibTex RIS Cite

Extensions of the operator Bellman and operator Holder type inequalities

Year 2024, , 12 - 29, 15.03.2024
https://doi.org/10.33205/cma.1435944

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.

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.
Year 2024, , 12 - 29, 15.03.2024
https://doi.org/10.33205/cma.1435944

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.
There are 25 citations in total.

Details

Primary Language English
Subjects Operator Algebras and Functional Analysis
Journal Section Articles
Authors

Mojtaba Bakherad 0000-0003-0323-6310

Fuad Kıttaneh 0000-0003-0308-365X

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

Cite

APA Bakherad, M., & Kıttaneh, F. (2024). Extensions of the operator Bellman and operator Holder type inequalities. Constructive Mathematical Analysis, 7(1), 12-29. https://doi.org/10.33205/cma.1435944
AMA Bakherad M, Kıttaneh F. Extensions of the operator Bellman and operator Holder type inequalities. CMA. March 2024;7(1):12-29. doi:10.33205/cma.1435944
Chicago Bakherad, Mojtaba, and Fuad Kıttaneh. “Extensions of the Operator Bellman and Operator Holder Type Inequalities”. Constructive Mathematical Analysis 7, no. 1 (March 2024): 12-29. https://doi.org/10.33205/cma.1435944.
EndNote Bakherad M, Kıttaneh F (March 1, 2024) Extensions of the operator Bellman and operator Holder type inequalities. Constructive Mathematical Analysis 7 1 12–29.
IEEE M. Bakherad and F. Kıttaneh, “Extensions of the operator Bellman and operator Holder type inequalities”, CMA, vol. 7, no. 1, pp. 12–29, 2024, doi: 10.33205/cma.1435944.
ISNAD Bakherad, Mojtaba - Kıttaneh, Fuad. “Extensions of the Operator Bellman and Operator Holder Type Inequalities”. Constructive Mathematical Analysis 7/1 (March 2024), 12-29. https://doi.org/10.33205/cma.1435944.
JAMA Bakherad M, Kıttaneh F. Extensions of the operator Bellman and operator Holder type inequalities. CMA. 2024;7:12–29.
MLA Bakherad, Mojtaba and Fuad Kıttaneh. “Extensions of the Operator Bellman and Operator Holder Type Inequalities”. Constructive Mathematical Analysis, vol. 7, no. 1, 2024, pp. 12-29, doi:10.33205/cma.1435944.
Vancouver Bakherad M, Kıttaneh F. Extensions of the operator Bellman and operator Holder type inequalities. CMA. 2024;7(1):12-29.