As applications, we present operator Grüss, Diaz–Metcalf, and Klamkin–McLenaghan inequalities.****************************************************************************************************************************************************************************************************************************************************
📌 3 Key Takeaways
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Objective & Hypothesis:In this paper, we present generalized Pólya-Szegö type inequalities for positive invertible operators on a Hilbert space for arbitrary operator means between the arithmetic and the harmonic means.
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Key Findings & Data:In this paper, we present generalized Pólya-Szegö type inequalities for positive invertible operators on a Hilbert space for arbitrary operator means between the arithmetic and the harmonic means.
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Conclusion & Impact:As applications, we present operator Grüss, Diaz–Metcalf, and Klamkin–McLenaghan inequalities.****************************************************************************************************************************************************************************************************************************************************
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Operator inequality · operator mean · positive linear map
Abstract
In this paper, we present generalized Pólya-Szegö type inequalities for positive invertible operators on a Hilbert space for arbitrary operator means between the arithmetic and the harmonic means. As applications, we present operator Grüss, Diaz–Metcalf, and Klamkin–McLenaghan inequalities.****************************************************************************************************************************************************************************************************************************************************
Dinh, T. H., Moradi, H. R., Sababheh, M. (2022). On the Pólya-Szegö operator inequality. Hacettepe Journal of Mathematics and Statistics, Cilt: 49 - Sayı: 5, 1744–1752. https://ajindex.com/articles/530062
T. Ando, Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear Algebra Appl. 26, 203–241, 1979
T. Ando and F. Hiai, Operator log-convex functions and operator means, Math. Ann. 350 (3), 611–630, 2011
J.C. Bourin, E.Y. Lee, M. Fujii and Y. Seo, A matrix reverse Hölder inequality, Linear Algebra Appl. 431 (11), 2154–2159, 2009
M. Fujii, S. Izumino, R. Nakamoto and Y. Seo, Operator inequalities related to Cauchy-Schwarz and Hölder-McCarthy inequalities, Nihonkai Math. J. 8, 117–122, 1997
J.I. Fujii, M. Nakamura, J. Pečarić and Y. Seo, Bounds for the ratio and difference between parallel sum and series via Mond-Pečarić method, Math. Inequal. Appl. 9 (4), 749–759, 2006
S. Furichi, H.R. Moradi and M. Sababheh, New sharp inequalities for operator means, Linear Multilinear Algebra, 67, 1567–1578, 2019
M.B. Ghaemi and V. Kaleibary, Some inequalities involving operator monotone func- tions and operator means, Math. Inequal. Appl. 19 (2), 757–764, 2016
I. Gumus, H.R. Moradi and M. Sababheh, More accurate operator means inequalities, J. Math. Anal. Appl. 465, 267–280, 2018
T.H. Dinh, T.H.B. Du and M.T. Ho On some matrix mean inequalities with Kan- torovich constant, Sci. Math. Jpn. 80 (2), 139–151, 2017
T.H. Dinh, M.S. Moslehian, C. Conde and P. Zhang, An extension of the Pólya-Szegö operator inequality, Expo. Math. 35 (2), 212–220, 2017
F. Kubo and T. Ando, Means of positive linear operators, Math. Ann. 246, 205–224, 1980
E.Y. Lee, A matrix reverse Cauchy-Schwarz inequality, Linear Algebra Appl. 430, 805–810, 2009
T. Furuta, J. Mićić, J. Pečarić and Y. Seo, Mond–Pečarić method in operator inequal- ities, Element, Zagreb, 2005
W. Specht, Zur Theorie der elementaren Mittel, Math. Z. 74, 91–98, 1960.