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Positivity

, Volume 22, Issue 5, pp 1311–1324 | Cite as

Norm inequalities related to the arithmetic–geometric mean inequalities for positive semidefinite matrices

  • Mostafa Hayajneh
  • Saja Hayajneh
  • Fuad Kittaneh
Article

Abstract

In this paper, we propose three new matrix versions of the arithmetic–geometric mean inequality for unitarily invariant norms, which stem from the fact that the Heinz mean of two positive real numbers interpolates between the geometric and arithmetic means of these numbers. Related trace inequalities are also presented.

Keywords

Unitarily invariant norm Hilbert–Schmidt norm Singular value Trace Positive semidefinite matrix Inequality 

Mathematics Subject Classification

Primary 15A60 Secondary 15A18 15A42 47A30 47B15 

References

  1. 1.
    Alakhrass, M.: Norm inequalities for sums and differences of products of matrices (preprint)Google Scholar
  2. 2.
    Ando, T., Zhan, X.: Norm inequalities related to operator monotone functions. Math. Ann. 315, 771–780 (1999)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ando, T., Hiai, F., Okubo, K.: Trace inequalities for multiple products of two matrices. Math. Inequal. Appl. 3, 307–318 (2000)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Audenaert, K.: A norm inequality for pairs of commuting positive semidefinite matrices. Electron. J. Linear Algebra 30, 80–84 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bhatia, R.: Matrix Analysis. Springer, New York (1997)CrossRefGoogle Scholar
  6. 6.
    Bhatia, R.: Trace inequalities for products of positive definite matrices. J. Math. Phys. 55, 013509 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bhatia, R., Davis, C.: More matrix forms of the arithmetic–geometric mean inequality. SIAM J. Matrix Anal. 14, 132–136 (1993)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bottazzi, T., Elencwajg, R., Larotonda, G., Varela, A.: Inequalities related to Bourin and Heinz means with a complex parameter. J. Math. Anal. Appl. 426, 765–773 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bourin, J.C.: Matrix versions of some classical inequalities. Linear Algebra Appl. 416, 890–907 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bourin, J.C.: Matrix subadditivity inequalities and block-matrices. Int. J. Math. 20, 679–691 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hayajneh, S., Kittaneh, F.: Lieb–Thirring trace inequalities and a question of Bourin. J. Math. Phys. 54, 033504 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hayajneh, S., Kittaneh, F.: Trace inequalities and a question of Bourin. Bull. Aust. Math. Soc. 88, 384–389 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hayajneh, M., Hayajneh, S., Kittaneh, F.: Remarks on some norm inequalities for positive semidefinite matrices and questions of Bourin. Math. Inequal. Appl. 20, 225–232 (2017)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Hayajneh, M., Hayajneh, S., Kittaneh, F.: On the Ando–Hiai–Okubo trace inequality. J. Oper. Theory 77, 77–86 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hayajneh, M., Hayajneh, S., Kittaneh, F.: Norm inequalities for positive semidefinite matrices and a question of Bourin. Int. J. Math. 28, 1750102 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Heinz, E.: Beitrage zur Storungstheorie der Spektralzerlegung. Math. Ann. 123, 415–438 (1951)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hoa, D.T.: An inequality for $t$-geometric means. Math. Inequal. Appl. 19, 765–768 (2016)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Kittaneh, F.: A note on the arithmetic–geometric mean inequality for matrices. Linear Algebra Appl. 171, 1–8 (1992)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lin, M.: Remarks on two recent results of Audenaert. Linear Algebra Appl. 489, 24–29 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Ltu, J.-T., Wang, Q.-W., Sun, F.-F.: On Hayajneh and Kittaneh’s conjecture on unitarily invariant norm. J. Math. Inequal. 11, 1019–1022 (2017)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Plevnik, L.: On a matrix trace inequality due to Ando, Hiai and Okubo. Indian J. Pure Appl. Math. 47, 491–500 (2016)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Simon, B.: Trace Ideals and Their Applications, Mathematical Surveys and Monographs, vol. 120, 2nd edn. American Mathematical Society, Providence (2005)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Mostafa Hayajneh
    • 1
  • Saja Hayajneh
    • 2
  • Fuad Kittaneh
    • 2
  1. 1.Department of MathematicsYarmouk UniversityIrbidJordan
  2. 2.Department of MathematicsThe University of JordanAmmanJordan

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