Skip to main content

Advertisement

SpringerLink
Log in

Further refinements of Young’s type inequality for positive linear maps

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In this work, we give a multiple-term refinement of Young’s inequality which allow us to generalize and unify several results. As applications, we provide further refinements of a reversed AM–GM operator inequalities which extends and unifies two recent and important results due to Yang et al. (Math Slovaca 69:919–930, 2019) and Ren et al. (J Inequal Appl 2020:98, 2020) for positive linear maps and matrices. Also, our work deals with several other related results to both scalar and operator versions of the generalized Young’s inequality. In particular, we give a multiple-term refinement of Young’s inequalities for Hilbert–Schmidt norm, the determinants and the traces of positive definite matrices.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alzer, H., da Fonseca, C.M., Kovacec, A.: Young-type inequalities and their matrix analogues. Linear Multilinear Algebra. 63(3), 622–635 (2015)

    Article  MathSciNet  Google Scholar 

  2. Ando, T.: Matrix Young inequality. Oper. Theory Adv. Appl. 75, 33–38 (1995)

    MATH  Google Scholar 

  3. Ando, T., Zhan, X.: Norm inequalities related to operator monotone functions. Math. Ann. 315, 771–780 (1999)

    Article  MathSciNet  Google Scholar 

  4. Bakherad, M.: Refinements of a reversed AM–GM operator inequality. Linear Multilinear Algebra 64, 1687–1695 (2016)

    Article  MathSciNet  Google Scholar 

  5. Bhatia, R.: Positive definite matrices. Princeton University Press, Princeton (2007)

    MATH  Google Scholar 

  6. Bhatia, R., Kittaneh, F.: Notes on matrix arithmetic-geometric mean inequalities. Linear Algebra Appl. 308, 203–211 (2000)

    Article  MathSciNet  Google Scholar 

  7. Burqan, A., Khandaqji, M.: Reverses of Young type inequalities. J. Math. Inequal. 9(1), 113–120 (2015)

    Article  MathSciNet  Google Scholar 

  8. Choi, D.: Multiple-term refinements of Young type inequalities. Hindawi J. Math. (2016)

  9. Fu, X.H., He, C.J.: Some operator inequalities for positive linear maps. Linear Multilinear Algebra 63, 571–577 (2015)

    Article  MathSciNet  Google Scholar 

  10. Furuichi, S.: Further improvements of Young inequality. Rev. R. Acad. Cienc. Exactas FÍs. Nat. Ser. A Mat. RACSAM. 113 (2019) 255–266. https://doi.org/10.1007/s13398-017-0469-5

  11. Horn, R.A., Johnson, C.R.: Matrix analysis. Cambridge Univ. Press, New-York (1985)

    Book  Google Scholar 

  12. Khan, S., Khan, M.A., Chu, Y.M.: New converses of Jensen inequality via Green functions with applications. Rev. R. Acad. Cienc. Exactas FÍs. Nat. Ser. A Mat. RACSAM. 114(3) (2020), Paper No. 114, 14 pp. https://doi.org/10.1007/s13398-020-00843-1

  13. Kittaneh, F.: Norm inequalities for fractionl powers of positive operators. Lett. Math. Phys. 27, 279–285 (1993)

    Article  MathSciNet  Google Scholar 

  14. Kórus, P.: A refinement of Young’s inequality. Acta Math. Hung. 153, 430–435 (2017)

    Article  MathSciNet  Google Scholar 

  15. Kubo, F., Ando, T.: Means of positive linear operators. Math. Ann. 246, 205–224 (1980)

    Article  MathSciNet  Google Scholar 

  16. Lin, M.: Squaring a reverse AM-+GM inequality. Stud. Math. 215, 187–194 (2013)

    Article  MathSciNet  Google Scholar 

  17. 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. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM. 113 (2019) 605–614. https://doi.org/10.1007/s13398-018-0499-7

  18. Pećarić, J., Furuta, T., Mićić, T., Seo, T.: Mond-Pećarić method in operator inequalities. Element, Zagreb (2005)

  19. Ren, Y., Li, P.: Further refinements of reversed AM-GM operator inequalities. J. Inequal. Appl. (2020)

  20. Ren, Y.: Some results of Young-type inequalities. Rev. R. Acad. Cienc. Exactas FÌs. Nat. Ser. A Mat. RACSAM 114(3) (2020), Paper No. 143, 10 pp. https://doi.org/10.1007/s13398-020-00880-w

  21. Sababheh, M., Choi, D.: A complete refinement of Young’s inequality. J. Math. Anal. Appl. 440(1): 379–393 (2016)

  22. Yang, C.J., Gao, Y.X., Lu, F.Y.: Some refinements of Young type inequality for positive linear map. Math. Slovaca 69, 919–930 (2019)

    Article  MathSciNet  Google Scholar 

  23. Yang, C.S., Li, Y.: Refinements and reverses of Young type inequalities. J. Math. Inequal. 14(2), 401–419 (2020)

    Article  MathSciNet  Google Scholar 

  24. Zhu, L.: New refinements of Young’s inequality. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM. 113 (2019) 909–915. https://doi.org/10.1007/s13398-018-0522-z

  25. Zhu, L.: Natural approachs of Young’s inequality. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM. 114(1) (2020), Paper No. 24, 11 pp. https://doi.org/10.1007/s13398-019-00770-w

Download references

Acknowledgements

The authors would like to express their deep thanks to the anonymous referees for their helpful comments and suggestions on the initial version of the manuscript which lead to the improvement of this paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculy of Sciences-Semlalia, University Cadi Ayyad, Av. Prince My. Abdellah, BP: 2390, 40000, Marrakesh, Marrakech, Morocco

    Mohamed Amine Ighachane & Mohamed Akkouchi

Authors
  1. Mohamed Amine Ighachane
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mohamed Akkouchi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohamed Amine Ighachane.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ighachane, M.A., Akkouchi, M. Further refinements of Young’s type inequality for positive linear maps. RACSAM 115, 94 (2021). https://doi.org/10.1007/s13398-021-01032-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13398-021-01032-4

Keywords

Mathematics Subject Classification

Search

Navigation