In this paper, some sharp bounds for Heinz mean by Heron mean and other means are presented. Further, we point out a mistake in [1] and correct it. Finally, we extend the results to the corresponding operator means.
Citation: Ling Zhu. Sharp bounds for Heinz mean by Heron mean and other means[J]. AIMS Mathematics, 2020, 5(1): 723-731. doi: 10.3934/math.2020049
In this paper, some sharp bounds for Heinz mean by Heron mean and other means are presented. Further, we point out a mistake in [1] and correct it. Finally, we extend the results to the corresponding operator means.
| [1] | G. Shi, Generalization of Heinz operator inequalities via hyperbolic functions, J. Math. Inequal., 13 (2019), 715-724. |
| [2] | R. Bhatia, C. Davis, More matrix forms of the arithmetic-geometric mean inequality, SIAM J. Matrix Anal. Appl., 14 (1993), 132-136. |
| [3] | R. Bhatia, Interpolating the arithmetic-geometric mean inequality and its operator version, Linear Algebra Appl., 413 (2006), 355-363. |
| [4] | F. Kittaneh, M. S. Moslehian, M. Sababheh, Quadratic interpolation of the Heinz means, Math. Inequal. Appl., 21 (2018), 739-757. |
| [5] | M. Sababheh, On the matrix harmonic mean, J. Math. Inequal., 12 (2018), 901-920. |
| [6] | M. Biernacki, J. Krzyz, On the monotonicity of certain functionals in the theory of analytic functions, Ann. Univ. M. Curie-Sklodowska, 2 (1955), 134-145. |
| [7] | F. Kittaneh, M. Krnic, Refined Heinz operator inequalities, Linear Multilinear A., 61 (2013), 1148-1157. |
| [8] | J. G. Zhao, J. L. Wu, H. S. Cao, et al. Operator inequalities involving the arithmetic, geometric, Heinz and Heron means, J. Math. Inequal., 8 (2014), 747-756. |
| [9] | M. Bakherad, M. S. Moslehian, Reverses and variations of Heinz inequality, Linear Multilinear A., 63 (2015), 1972-1980. |
| [10] | T. H. Dinh, R. Dumitru, J. A. Franco, The matrix power means and interpolations, Adv. Oper. Theory, 3 (2018), 647-654. |
| [11] | Y. Kapil, C. Conde, M. S. Moslehian, et al. Norm inequalities related to the Heron and Heinz means, Mediterr. J. Math., 14 (2017), 213. |
| [12] | M. Ito, Estimations of the Lehmer mean by the Heron mean and their generalizations involving refined Heinz operator inequalities, Adv. Oper. Theory, 3 (2018), 763-780. |
| [13] | M. Singh, Inequalities involving eigenvalues for difference of operator means, Electron. J. Linear AL., 27 (2014), 557-568. |
Ling Zhu. Sharp bounds for Heinz mean by Heron mean and other means[J]. AIMS Mathematics, 2020, 5(1): 723-731. doi: 10.3934/math.2020049
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