On the Pólya-Szegö operator inequality

Year 2020, Volume: 49 Issue: 5, 1744 - 1752, 06.10.2020

Trung Hoa Dinh Hamid Reza Moradı , Mohammad Sababheh

https://doi.org/10.15672/hujms.547158

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.

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Keywords

Operator inequality , operator mean , positive linear map , Pólya-Szegö inequality , operator monotone functions

References

• [1] T. Ando, Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear Algebra Appl. 26, 203–241, 1979.
• [2] T. Ando and F. Hiai, Operator log-convex functions and operator means, Math. Ann. 350 (3), 611–630, 2011.
• [3] J.C. Bourin, E.Y. Lee, M. Fujii and Y. Seo, A matrix reverse Hölder inequality, Linear Algebra Appl. 431 (11), 2154–2159, 2009.
• [4] 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.
• [5] 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.
• [6] S. Furichi, H.R. Moradi and M. Sababheh, New sharp inequalities for operator means, Linear Multilinear Algebra, 67, 1567–1578, 2019.
• [7] M.B. Ghaemi and V. Kaleibary, Some inequalities involving operator monotone func- tions and operator means, Math. Inequal. Appl. 19 (2), 757–764, 2016.
• [8] I. Gumus, H.R. Moradi and M. Sababheh, More accurate operator means inequalities, J. Math. Anal. Appl. 465, 267–280, 2018.
• [9] 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.
• [10] 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.
• [11] F. Kubo and T. Ando, Means of positive linear operators, Math. Ann. 246, 205–224, 1980.
• [12] E.Y. Lee, A matrix reverse Cauchy-Schwarz inequality, Linear Algebra Appl. 430, 805–810, 2009.
• [13] T. Furuta, J. Mićić, J. Pečarić and Y. Seo, Mond–Pečarić method in operator inequal- ities, Element, Zagreb, 2005.
• [14] W. Specht, Zur Theorie der elementaren Mittel, Math. Z. 74, 91–98, 1960 .