Abstract
Some Choi–Davis–Jensen’s type trace inequalities for convex functions are proved. Also, we generalize these inequalities for any arbitrary operator mean via operator monotone decreasing functions. In particular, we present some new order among \(\mathrm{tr}(\Phi (C)A)\) and \(\mathrm{tr}(\Phi (C)A^{-1})\). New refinements of some power type trace inequalities via reverse and refinement of Young’s inequality are established. Among our results, we obtain new versions of the Hölder type trace inequality for any arbitrary operator mean.
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References
Ando, T.: Concavity of certain maps on positive definite matrices and applications to Hadamard products. linear Algebra Appl. 26, 203–241 (1979)
Ando, T., Zhan, X.: Norm inequalities related to operator monoton functions. Math. Ann. 315, 771–780 (1999)
Ando, T., Hiai, F.: Operator log-convex functions and operator means. Math. Ann 350(3), 611–630 (2011)
Bellman, R.: Some inequalities for positive definite matrices. In: Beckenbach, E.F. (ed.) General Inequalities 2, Proceedings of the 2nd International Conference on General Inequalities, pp. 89–90. Birkhauser, Basel (1980)
Bhatia, R.: Positive Definite Matrices. Princeton University Press, Princeton (2007)
Blanchard, P., Brüning, E.: Mathematical Methods in Physics. Springer, Switzerland (2015)
Choi, M.D.: A Schwarz inequality for positive linear maps on \(C^*\)-algebras. Ill. J. Math. 18, 565–574 (1974)
Coop, I.D.: On matrix trace inequalities and related topics for products of Hermitian matrix. J. Math. Anal. Appl. 188, 999–1001 (1994)
Davis, C.: A Schwarz inequality for convex operator functions. Proc. Am. Math. Soc. 8, 42–44 (1957)
Dragomir, S.S.: On some Hölder type trace inequalities for operator weighted geometric mean. RGMIA Res. Rep. Coll. 18, Art. 152 (2015)
Dragomir, S.S.: Jensen’s type trace inequalities for convex functions of selfadjoint operators in Hilbert spaces. Facta Univ. Ser. Math. Inform. 31(5), 981–998 (2016)
Dragomir, S.S.: Reverse Jensen’s type trace inequalities for convex functions of selfadjoint operators in Hilbert spaces. Ann. Math. Sil. 30(1), 39–62 (2016)
Dragomir, S.S.: Some Slater’s type trace inequalities for convex functions of selfadjoint operators in Hilbert spaces. Toyama Math. J. 38, 75–99 (2016)
Dragomir, S.S.: Some trace inequalities for convex functions of selfadjoint operators in Hilbert spaces. Korean J. Math. 24(2), 273–296 (2016)
Dragomir, S.S.: Power and Hölder type trace inequalities for positive operators in Hilbert spaces. Stud. Sci. Math. Hungar. 55(3), 383–406 (2018)
Furuichi, Sh, Moradi, H.R., Sababheh, M.: New sharp inequalities for operator means. Linear Multilinear Algebra 66(1), 1–12 (2018)
Khosravi, M., Moslehian, M.S., Sheikhhosseini, A.: Some operator inequalities involving operator means and positive linear maps. Linear Multilinear Algebra 66(6), 1186–1198 (2018)
Liao, W., Wu, J., Zhao, J.: New versions of reverse Young and Heinz mean inequalities with the Kantorovich constant. Taiwan. J. Math. 19(2), 467–479 (2015)
Moradi, H.R., Furuichi, S., Mitroi-Symeonidis, F.-C., Naseri, R.: An extension of Jensen’s operator inequality and its application to Young inequality. Rev. Real Acad. Cie. Exactas Fís. Nat. Ser. A. Mat. 113, 605–614 (2019)
Neudecker, H.: A matrix trace inequality. J. Math. Anal. Appl. 166, 302–303 (1992)
Ruskai, M.B.R.: Inequalities for traces on Von Neumann algebras. Commun. Math. Phys. 26, 280–289 (1972)
Simon, B.: Trace Ideals and Their Applications. Cambridge University Press, Cambridge (1979)
Stormer, E.: Positive Linear Maps of Operator Algebras. Springer, Berlin (2013)
Yang, Y.: A matrix trace inequality. J. Math. Anal. Appl. 133, 573–574 (1988)
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Jafarmanesh, H., Shateri, T.L. & Dragomir, S.S. Choi–Davis–Jensen’s type trace inequalities for convex functions of self-adjoint operators in Hilbert spaces. Bol. Soc. Mat. Mex. 26, 1195–1215 (2020). https://doi.org/10.1007/s40590-020-00300-4
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DOI: https://doi.org/10.1007/s40590-020-00300-4