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  • 1 Victoria University, PO Box 14428, Melbourne City, MC 8001, Australia
  • | 2 University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050, South Africa
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In this paper we obtain some new power and hölder type trace inequalities for positive operators in Hilbert spaces. As tools, we use some recent reverses and refinements of Young inequality obtained by several authors.

  • [1]

    Ando, T., Matrix Young inequalities, Oper. Theory Adv. Appl., 75 (1995), 33-38.

  • [2]

    Bellman, R., Some inequalities for positive definite matrices, in: E. F. Beckenbach (Ed.), General Inequalities 2, Proceedings of the 2nd International Conference on General Inequalities, Birkhäuser, Basel, 1980, pp. 89-90.

    • Search Google Scholar
    • Export Citation
  • [3]

    Belmega, E. V., Jungers, M. and Lasaulce, S., A generalization of a trace inequality for positive definite matrices, Aust. J. Math. Anal. Appl., 7(2) (2010), Art. 26, 5 pp.

    • Search Google Scholar
    • Export Citation
  • [4]

    Chang, D., A matrix trace inequality for products of Hermitian matrices, J. Math. Anal. Appl., 237 (1999), 721-725.

  • [5]

    Chen, L. and Wong, C., Inequalities for singular values and traces, Linear Algebra Appl., 171 (1992), 109-120.

  • [6]

    Coop, I. D., On matrix trace inequalities and related topics for products of Hermitian Matrix, J. Math. Anal. Appl., 188 (1994) 999-1001.

    • Search Google Scholar
    • Export Citation
  • [7]

    Dragomir, S.S., Bounds for the normalized Jensen functional, Bull. Austral. Math. Soc., 74(3) (2006), 417-478.

  • [8]

    Dragomir, S. S., Some Trace Inequalities of Čebyšev Type for Functions of Operators in Hilbert Spaces, Linear Multilinear Algebra, 64 (2016), no. 9, 1800-1813. Preprint RGMIA Res. Rep. Coll., 17 (2015), Art. 111. [Online http://rgmia.org/papers/v17/v17a111.pdf].

    • Search Google Scholar
    • Export Citation
  • [9]

    Dragomir, S. S., Some GrUss type inequalities for trace of operators in Hilbert spaces, Oper. Matrices, 10 (2016), no. 4, 923-943. Preprint RGMIA Res. Rep. Coll., 17 (2015), Art. 114. [Online http://rgmia.org/papers/v17/v17a114.pdf].

    • Search Google Scholar
    • Export Citation
  • [10]

    Dragomir, S. S., A note on Young's inequality, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM, 111 (2017), no. 2, 349-350. Preprint RGMIA Res. Rep. Coll., 18 (2015), Art. 126. [http://rgmia.org/papers/v18/v18a126.pdf].

  • [11]

    Dragomir, S. S., A note on new refinements and reverses of Young's inequality, Transylv. J. Math. Mech., 8 (2016), no. 1, 45-49. Preprint RGMIA Res. Rep. Coll., 18(2015), Art. 131. [Online http://rgmia.org/papers/v18/v18a131.pdf].

    • Search Google Scholar
    • Export Citation
  • [12]

    Dragomir, S. S., On some Hölder type trace inequalities for operator weighted geometric mean,

  • [13]

    Dragomir, S. S. and Mcandrew, A., A note on numerical comparison of some multiplicative bounds related to weighted arithmetic and geometric mean, Preprint RGMIA Res. Rep. Coll., 19 (2016), Art. 102. 10 pp. [Online http://rgmia.org/papers/v19/v19a102.pdf].

  • [14]

    Fuji, M., Izumino, S., Nakamoto, R. and Seo, Y., Operator inequalities related to Cauchy-Schwarz and Holder-McCarthy inequalities, Nihonkai Math. J., 8 (1997), 117-122.

    • Search Google Scholar
    • Export Citation
  • [15]

    Furuichi, S. and Lin, M., Refinements of the trace inequality of Belmega, Lasaulce and Debbah, Aust. J. Math. Anal. Appl., 7(2) (2010), Art. 23, 4 pp.

    • Search Google Scholar
    • Export Citation
  • [16]

    Furuichi, S. and Minculete, N., Alternative reverse inequalities for Young's Inequality, J. Math Inequal., 5(4) (2011), 595-600.

  • [17]

    Furuta, T., Hot, J. M., Pecaric, J. and Seo, Y., Mond-Pecaric Method in Operator Inequalities. Inequalities for Bounded Selfadjoint Operators on a Hilbert Space, Element, Zagreb, 2005.

  • [18]

    Greub, W. and Rheinboldt, W., On a generalisation of an inequality of L. V. Kantorovich, Proc. Amer. Math. Soc., 10 (1959), 407-415.

  • [19]

    Kittaneh, F. and Manasrah, Y., Improved Young and Heinz inequalities for Matrix, J. Math. Anal. Appl., 361 (2010), 262-269.

  • [20]

    Kittaneh, F. and Manasrah, Y., Reverse Young and Heinz inequalities for matrices, Linear Multilinear Algebra., 59 (2011), 1031-1037.

  • [21]

    Lee, H. D., On some matrix inequalities, Korean J. Math., 16(4) (2008), 565-571.

  • [22]

    Liao, W., Wu, J. and Zhao, J., New versions of reverse Young and Heinz mean inequalities with the Kantorovich constant, Taiwanese J. Math., 19(2) (2015), 467-479.

    • Search Google Scholar
    • Export Citation
  • [23]

    Liu, L., A trace class operator inequality, J. Math. Anal. Appl., 328 (2007), 1484-1486.

  • [24]

    Manjegani, S., Holder and Young inequalities for the trace of operators, Positivity 11 (2007), 239-250.

  • [25]

    Neudecker, H., A matrix trace inequality, J. Math. Anal. Appl., 166 (1992) 302-303.

  • [26]

    Ruskai, M. B., Inequalities for traces on von Neumann algebras, Commun. Math. Phys., 26 (1972), 280289.

  • [27]

    Shebrawi, K. and Albadawi, H., Operator norm inequalities of Minkowski type, J. Inequal. Pure Appl. Math., 9(1) (2008), 1-10, article 26.

    • Search Google Scholar
    • Export Citation
  • [28]

    Shebrawi, K. and Albadawi, H., Trace inequalities for matrices, Bull. Aust. Math. Soc., 87 (2013), 139-148.

  • [29]

    Shisha, O. and Mond, B., Bounds on differences of means, Inequalities I, New York-London, 1967, 293-308.

  • [30]

    Simon, B., Trace Ideals and Their Applications, Cambridge University Press, Cambridge, 1979.

  • [31]

    Specht, W., Zer Theorie der elementaren Mittel, Math. Z., 74 (1960), 91-98.

  • [32]

    Tominaga, M., Specht's ratio in the Young inequality, Sci. Math. Japon., 55 (2002), 583-588.

  • [33]

    Ulukök, Z. and Turkmen, R., On some matrix trace inequalities, J. Inequal. Appl., 2010, Art. ID 201486, 8 pp.

  • [34]

    Yang, X., A matrix trace inequality, J. Math. Anal. Appl., 250 (2000), 372-374.

  • [35]

    Yang, X. M., Yang, X. Q. and Teo, K. L., A matrix trace inequality, J. Math. Anal. Appl., 263 (2001), 327-331.

  • [36]

    Yang, Y., A matrix trace inequality, J. Math. Anal. Appl., 133 (1988), 573-574.

  • [37]

    Zuo, G., Shi, G. and Fujii, M., Refined Young inequality with Kantorovich constant, J. Math. Inequal., 5 (2011), 551-556.