Abstract
In this paper, by virtue of an expression of matrix geometric means for positive semidefinite matrices via the Moore-Penrose inverse, we show matrix versions of the Hölder-McCarthy inequality, the Hölder inequality and quasi-arithmetic power means via matrix geometric means, and their reverses for positive definite matrices via the generalized Kantorovich constant.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bhatia, R.: Positive Definite Matrices, Princeton Series and Applied Mathematics. Princeton University Press, Princeton (2007)
Bhatia, R., Davis, C.: More operator versions of the Schwarz inequality. Commun. Math. Phys. 215, 239–244 (2000)
Drury, S.W., Liu, S., Lu, C.Y., Puntanen, S., Styan, G.: Some comments on several matrix inequalities with applications to canonical correlations: historical backgroud and recent developments. Indian J. Stat. 64, 453–507 (2002)
Fan, Ky: Some matrix inequalities. Abh. Math. Sem. Univ. Hamburg 29, 185–196 (1966)
Fujii, M., Mićić Hot, J., Pečarić, J., Seo, Y.: Recent developments of Mond-Pečarić method in operator inequalities, monographs in inequalities vol 4, Element, Zagreb (2012)
Fujii, J.I.: Operator-valued inner product and operator inequalities. Banach J. Math. Anal. 2, 59–67 (2008)
Fujii, J.I.: Moore-Penrose inverse and operator mean. Sci. Math. Jpn. 82(2), 125–129 (2019)
Fujii, J.I., Seo, Y.: Tsallis relative operator entropy with negative paramaeters. Adv. Op. Theory 1(2), 219–236 (2016)
Fujimoto, M., Seo, Y.: Matrix Wielandt inequality via the matrix geometric mean. Linear Multilinear Algebra 66, 1564–1577 (2018)
Fujimoto, M., Seo, Y.: The Schwarz inequality via operator-valued inner product and the geometric operator mean. Linear Algebra Appl. 561, 141–160 (2019)
Furuta, T., Mićić Hot, J., Pečarić, J., Seo, Y.: Mond-Pečarić method in Operator Inequalities, Monographs in Inequalities 1, Elements, Zagreb (2005)
Furuta, T.: Operator inequalities associated with Holder-McCarthy and Kantorovich inequalities. J. Inequal. Appl. 2, 137–148 (1998)
Greub, W., Rheinboldt, W.: On a generalization of an inequality of L.V. Kantorovich. Proc. Am. Math. Soc. 10, 407–415 (1959)
Hiai, F., Petz, D.: Introduction to Matrix Anaysis and Applications. Springer, Berlin (2014)
Kantorovich, L.V.: Functional analysis and applied mathematics, Uspehi Mat. Nauk., 3, pp.89-185. Translated from the Russian by Curtis D. Benster, National Bureau of Standards, Report 1509, March 7, 1952 (1948)
Kubo, F., Ando, T.: Means of positive linear operators. Math. Ann. 246, 205–224 (1980)
Marshall, A.W., Olkin, I.: Matrix versions of the Cauchy and Kantorovich inequalities. Aequationes Math. 40, 89–93 (1990)
Ohya, M., Petz, D.: Quantum Entropy and its Use. Springer, Berlin (1993)
Pusz, W., Woronowicz, S.L.: Functional calculus for sesquilinear forms and the purification map. Rep. Math. Phys. 8, 159–170 (1975)
Specht, W.: Zur Theorie der elementaren Mittel. Math. Z. 74, 91–98 (1960)
Turing, A.M.: Rounding off-errors in matrix processes. Quart. J. Mech. Appl. Math. 1, 287–308 (1948)
Werner, H.J.: On the matrix monotonicity of generalized inversion. Linea Algebra Appl. 27, 141–145 (1979)
Acknowledgements
The second author is supported by Grant-in-Aid for Scientific Research (C), JSPS KAKENHI Grant no. JP 19K03542.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Hiroyuki Osaka.
Dedicated to honor of Professor Rajendra Bhatia
Rights and permissions
About this article
Cite this article
Nakayama, R., Seo, Y. & Tojo, R. Matrix Hölder-McCarthy inequality via matrix geometric means. Adv. Oper. Theory 5, 744–767 (2020). https://doi.org/10.1007/s43036-020-00044-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s43036-020-00044-y