Abstract
Let \(P_t(\mu )\) be the matrix power mean of a compactly supported probability measure \(\mu \) on the set of positive definite matrices \(\mathbb {P}_n\). We present an inequality involving the matrix power mean \(P_t(\mu )\) and its adjoint \(P_{-t}(\mu )\). Our result gives in particular a comparison of the quasi arithmetic mean \(\sharp _{\omega ,t}\) and its adjoint \(\sharp _{\omega ,-t}\).
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References
Bhatia, R.: Positive Definite Matrices. Princeton University Press, Princeton (2007)
Bhatia, R., Kittaneh, F.: Notes on matrix arithmetic–geometric mean inequalities. Linear Algebra Appl. 308, 203–211 (2000)
Dehghani, M., Kian, M., Seo, Y.: Developed matrix inequalities via positive multilinear mappings. Linear Algebra Appl. 484, 63–85 (2015)
Dehghani, M., Kian, M., Seo, Y.: Matrix power mean and the information monotonicity. Linear Algebra Appl. 521, 57–69 (2017)
Fujii, J.I., Seo, Y.: The unique solution of the Karcher equation and the self-adjointness of the Karcher mean. Linear Multilinear Algebra 67, 976–986 (2019)
Kamei, E.: Geometrical expansion of an operator equation. Sci. Math. Jpn. 80, 153–160 (2017)
Khosravi, M., Moslehian, M.S., Sheikhhosseini, A.: Some operator inequalities involving operator means and positive linear maps. Linear Multilinear Algebra 66, 1186–1198 (2018)
Kian, M., Moslehian, M.S.: Power means of probability measures and Ando-Hiai inequality. arXiv:1806.04210v2 [math.FA]
Kian, M., Moslehian, M.S., Seo, Y.: Variants of Ando-Hiai inequality for operator power means. Linear Multilinear Algebra (2019). https://doi.org/10.1080/03081087.2019.1635981
Kim, S., Lee, H.: The power mean and the least squares mean of probability measures on the space of positive definite matrices. Linear Algebra Appl. 465, 325–346 (2015)
Kudo, F., Ando, T.: Means of positive linear operators. Math. Ann. 246, 205–224 (1980)
Lawson, J., Lim, Y.: Karcher means and Karcher equations of positive definite operators. Trans. Am. Math. Soc. Ser. B 1, 1–22 (2014)
Lim, Y., Pálfia, M.: Matrix power means and the Karcher mean. J. Funct. Anal. 262, 1498–1514 (2012)
Seo, Y.: Operator power means due to Lawson–Lim–Palfia for \(1<t<2\). Linear Algebra Appl. 459, 342–356 (2014)
Acknowledgements
The author is grateful to Dr. Mohammad Mehrpooya for his helps. This research was in part supported by a Grant from University of Bojnord (No. 97/367/6055).
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Communicated by Hamid Reza Ebrahimi Vishki.
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Kian, M. Some Inequalities for Matrix Power Means. Bull. Iran. Math. Soc. 46, 893–903 (2020). https://doi.org/10.1007/s41980-019-00299-z
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DOI: https://doi.org/10.1007/s41980-019-00299-z