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Results in Mathematics

, 74:16 | Cite as

Reverses of Young Type Inequalities for Matrices Using the Classical Kantorovich Constant

  • Leila NasiriEmail author
  • Mahmood Shakoori
Article
  • 55 Downloads

Abstract

In this article, we give some reverses of Young type inequalities which were established by Burqan and Khandaqji (J Math Inequal 9:113–120, 2015) applying the Kantorovich constant. As an application of these numerical versions, we study some matrix inequalities for the Hilbert–Schmidt norm and the trace norm.

Keywords

Hilbert–Schmidt norm reverse Young type inequality classical Kantorovich constant matrix inequalities 

Mathematics Subject Classification

47A30 15A45 15A60 

Notes

Acknowledgements

The authors would like to thank the handling editor and referees for giving valuable comments and suggestions to improve our manuscript. L. Nasiri (the corresponding author) and M. Shakoori would like to thank the Lorestan University.

References

  1. 1.
    Ando, T.: Matrix Young inequality. Oper. Theory Adv. Appl. 75, 33–38 (1995)zbMATHGoogle Scholar
  2. 2.
    Burqan, A., Khandaqji, M.: Reverses of Young type inequalities. J. Math. Inequal. 9, 113–120 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bhatia, R.: Matrix Analysis. Springer, New York (1997)CrossRefGoogle Scholar
  4. 4.
    Bhatia, R., Parthasarathy, K.R.: Positive definite functions and operators inequalities. Bull. Lond. Math. Soc. 32, 214–228 (2000)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bhatia, R., Kittaneh, F.: Notes on matrix arithmetic-geometric mean inequalities. Linear Algebra Appl. 308, 203–211 (2000)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bakherad, M., Krnic, M., Moslehian, M.S.: Reverses of the Young inequality for matrices and operators. Rocky Mt. J. Math. 46, 1089–1105 (2016)CrossRefGoogle Scholar
  7. 7.
    Bhatia, R., Kittaneh, F.: On singular values of a product of operators. SIAM J. Matrix Anal. Appl. 11, 271–277 (1990)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Choi, D., Krnic, M., Pecaric, J.: Improved Jensen-type inequalities via linear interpolation and applications. J. Math. Inequal. 11, 301–322 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Cartwright, D., Field, M.: A refinement of the arithmetic mean-geometric mean inequality. Proc. Am. Math. Soc. 71, 36–38 (1978)MathSciNetCrossRefGoogle Scholar
  10. 10.
    He, C., Zou, L., Qaisar, S.: On improved arithmetic-geometric mean and Heinz inequalities for matrices. J. Math. Inequal. 6, 453–459 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hu, X., Xue, J.: A note on reverses of Young type inequalities. J. Inequal. Appl. 2015, 98 (2015).  https://doi.org/10.1186/sB660-015-0622-7 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hirzallah, O., Kittaneh, F.: Matrix Young inequalities for the Hilbert–Schmidt norm. Linear Algebra Appl. 308, 77–84 (2000)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hu, X.: Young type inequalities for matrices. J. East China Normal Univ. Nat. Sci. 4, 12–17 (2012). Article ID: 1000-5641(2012)04-0012-560MathSciNetGoogle Scholar
  14. 14.
    Kittaneh, F., Manasrah, Y.: Improved Young and Heinz inequalities for matrices. J. Math. Anal. Appl. 361, 262–269 (2010)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kittaneh, F.: On some operator inequalities. Linear Algebra Appl. 208(209), 19–28 (1994)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kittaneh, F., Manasrah, Y.: Reverse Young and Heinz inequalities for matrices. Linear Multilinear Algebra 59, 1031–1037 (2011)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kosaki, H.: Arithmetic-geometric mean and related inequalities for operators. J. Funct. Anal. 156, 429–451 (1998)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Liao, W., Wu, J., Zhao, J.: New versions of reverse Young and Heinz mean inequalities with the Kantorovich constant. Taiwan. J. Math. 19, 467–479 (2015).  https://doi.org/10.11650/tjm.19.2015.4548 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Nasiri, L., Shakoori, M.: A note on improved Young type inequalities with Kantorovich constant. J. Math. Stat. 12, 201–205 (2016)CrossRefGoogle Scholar
  20. 20.
    Nasiri, L., Shakoori, M., Liao, W.: A note on the Young type inequalities. Int. J. Nonlinear Appl. 10(2), 559–570 (2016)zbMATHGoogle Scholar
  21. 21.
    Nasiri, L., Liao, W.: The new reverses of Young type inequalities for numbers, matrices and operators. Oper. Matrices 12(4), 1063–1071 (2018)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Nasiri, L., Bakherad, M.: Improvements of some operator inequalities involving positive linear maps via the Kantorovich constant. Houston J. Math. (to appear)Google Scholar
  23. 23.
    Tominago, M.: Spechts ratio in the Young inequality. Sci. Math. Japon 55, 583–588 (2002)MathSciNetGoogle Scholar
  24. 24.
    Wu, J.L., Zhao, J.G.: Operator inequalities and reverse inequalities related to the Kittaneh–Manasrah inequalities. Linear Multilinear Algebra.  https://doi.org/10.1080/03081087.2013.794235
  25. 25.
    Zuo, H., Shi, G., Fujii, M.: Refined Young inequality with Kantorovich constant. J. Math. Inequal. 5, 551–556 (2011)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Zhan, X.: Inequalities for unitarily invariant norms. SIAM J. Matrix Anal. Appl. 3, 466–470 (1998)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Computer Science, Faculty of ScienceLorestan UniversityKhorramabadIran

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