New sharp inequalities for operator means. (English) Zbl 1502.47027
Summary: New sharp multiplicative reverses of the operator means inequalities are presented, with a simple discussion of squaring an operator inequality. As a direct consequence, we extend the operator Pólya-Szegő inequality to arbitrary operator means. Furthermore, we obtain some new lower and upper bounds for the Tsallis relative operator entropy, operator monotone functions and positive linear maps.
MSC:
| 47A64 | Operator means involving linear operators, shorted linear operators, etc. |
| 47A63 | Linear operator inequalities |
| 46L05 | General theory of \(C^*\)-algebras |
| 47A60 | Functional calculus for linear operators |
Keywords:
operator inequality; positive linear map; Pólya-Szegő inequality; operator monotone functionReferences:
| [1] | Bhatia, R., Matrix analysis (1997), New York (NY): Springer-Verlag, New York (NY) |
| [2] | Liao, W.; Wu, J.; Zhao, J., New versions of reverse Young and Heinz mean inequalities with the Kantorovich constant, Taiwanese J Math, 19, 2, 467-479 (2015) · Zbl 1357.26048 |
| [3] | Sababheh, M., Convexity and matrix means, Linear Algebra Appl, 506, 588-602 (2016) · Zbl 1346.15019 |
| [4] | Sababheh, M., Convex functions and means of matrices, Math Inequal Appl, 17, 29-47 (2017) · Zbl 1362.15018 |
| [5] | Zuo, H.; Shi, G.; Fujii, M., Refined Young inequality with Kantorovich constant, J Math Inequal, 5, 4, 551-556 (2011) · Zbl 1242.47017 |
| [6] | Lee, E-Y, A matrix reverse Cauchy-Schwarz inequality, Linear Algebra Appl, 430, 805-810 (2009) · Zbl 1157.15024 |
| [7] | Hoa, Dt; Moslehian, Ms; Conde, C., An extension of the Pólya-Szegö operator inequality, Expo Math, 35, 2, 212-220 (2017) · Zbl 1377.47007 |
| [8] | Fujii, Ji; Nakamura, M.; Pečarić, J., Bounds for the ratio and difference between parallel sum and series via Mond-Pečarić method, Math Inequal Appl, 9, 4, 749-759 (2006) · Zbl 1113.47007 |
| [9] | Gumus, Ih; Moradi, Hr; Sababheh, M., More accurate operator means inequalities · Zbl 1509.47022 |
| [10] | Lin, M., Squaring a reverse AM-GM inequality, Studia Math, 215, 187-194 (2013) · Zbl 1276.47022 |
| [11] | Lin, M., On an operator Kantorovich inequality for positive linear maps, J Math Anal Appl, 402, 127-132 (2013) · Zbl 1314.47024 |
| [12] | Fu, X.; He, C., Some operator inequalities for positive linear maps, Linear Multilinear Algebra, 63, 571-577 (2015) · Zbl 1322.47026 |
| [13] | Furuta, T., Invitation to linear operators from matrices to bounded linear operators on a Hilbert space (2001), London: Taylor & Francis, London · Zbl 1029.47001 |
| [14] | Furuichi, S.; Yanagi, K.; Kuriyama, K., A note on operator inequalities of Tsallis relative operator entropy, Linear Algebra Appl, 407, 19-31 (2005) · Zbl 1071.47021 |
| [15] | Bakherad, M.; Moslehian, Ms, Reverses and variations of Heinz inequality, Linear Multilinear Algebra, 63, 10, 1972-1980 (2015) · Zbl 1354.47017 |
| [16] | Ando, T.; Hiai, F., Operator log-convex functions and operator means, Math Ann, 350, 3, 611-630 (2011) · Zbl 1221.47028 |
| [17] | Ghaemi, Mb; Kaleibary, V., Some inequalities involving operator monotone functions and operator means, Math Inequal Appl, 19, 757-764 (2016) · Zbl 1372.47026 |
| [18] | Bourin, J-C; Lee, E-Y; Fujii, M., A matrix reverse Hölder inequality, Linear Algebra Appl, 431, 2154-2159 (2009) · Zbl 1179.15021 |
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