Skip to main content
SpringerLink
Log in

On some new generalizations of Hua’s inequality

  • Published:
Acta Mathematica Hungarica Aims and scope Submit manuscript

Abstract

We give an extension of Hua’s inequality for C*-valued norm in pre-Hilbert C*-modules setting. As its applications, some known and new generalizations of Hua’s inequality are deduced. In particular, we establish the connection between C*-valued norm triangle inequality and C*-valued norm Hua’s inequality. We also present some operator versions of Hua’s inequality on Hilbert spaces. As a consequence, we obtain various forms of Hua’s inequality. Moreover, we indicate that for our operator, Hua’s inequality is equivalent to operator convexity of a given continuous real function.

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

Reference

  1. Arambašić, L., Rajić, R.: On the \(C^*\)-valued triangle equality and inequality in Hilbert \(C^*\)-modules. Acta Math. Hungar. 119, 373–380 (2008)

    Article  MathSciNet  Google Scholar 

  2. Choi, M.D.: A Schwarz inequality for positive linear maps on \(C^{*}\)-algebras. Illinois J. Math. 18, 565–574 (1974)

    Article  MathSciNet  Google Scholar 

  3. Davis, C.: A Schwarz inequality for convex operator functions. Proc. Amer. Math. Soc. 8, 42–44 (1957)

    Article  MathSciNet  Google Scholar 

  4. Dragomir, S.S., Yang, G.S.: On Hua's inequality in real inner product spaces. Tamkang J. Math. 27, 227–232 (1996)

    MathSciNet  MATH  Google Scholar 

  5. Drnovsek, R.: An operator generalization of the Lo-Keng Hua inequality. J. Math. Anal. Appl. 196, 1135–1138 (1995)

    Article  MathSciNet  Google Scholar 

  6. T. Furuta, J. M. Hot, J. E. Pec̆arić, and Y. Seo, Mond–Pecaric Method in Operator Inequalities, Element (Zagreb, 2005)

  7. Gao, F.G., Hong, G.Q.: Generalizations of Hua's inequality in Hilbert \(C^*\)-modules. J. Comput. Anal. Appl. 26, 415–420 (2019)

    Google Scholar 

  8. Hansen, F., Pedersen, G.K.: Jensen's operator inequality. Bull. Lond. Math. Soc. 35, 553–564 (2003)

    Article  MathSciNet  Google Scholar 

  9. Hua, L.K.: Additive Theory of Prime Numbers. Amer. Math, Soc (1965)

    MATH  Google Scholar 

  10. Kaplansky, I.: Modules over operator algebras. Amer. J. Math. 75, 839–858 (1953)

    Article  MathSciNet  Google Scholar 

  11. Kolarec, B.: Inequalities for the \(C^*\)-valued norm on a Hilbert \(C^*\)-module. Math. Inequal. Appl. 12, 745–751 (2009)

    MathSciNet  MATH  Google Scholar 

  12. E.C. Lance, Hilbert \(C^{*}\)-modules: a Toolkit for Operator Algebraists, London Math. Soc. Lecture Note Series, 210, Cambridge University Press (1995)

  13. Murphy, G.J.: \(C^{*}\)-algebras and Operator Theory. Academic Press (1990)

  14. Moslehian, M.S.: Operator extensions of Hua's inequality. Linear Algebra Appl. 430, 1131–1139 (2009)

    Article  MathSciNet  Google Scholar 

  15. Moslehian, M.S., Fujii, J.I.: Operator inequalities related to weak 2-positivity. J. Math. Inequal. 7, 175–182 (2013)

    Article  MathSciNet  Google Scholar 

  16. Paschke, W.L.: Inner product modules over \(B^*\)-algebras. Trans. Amer. Math. Soc. 182, 443–468 (1973)

    MathSciNet  MATH  Google Scholar 

  17. V.I. Paulsen, Completely Bounded Maps and Dilations, Pitman Research Notes in Math. Ser., 146, Longman (1986)

  18. C. E. M. Pearce and J. E. Pec̆arić, A remark on the Lo-Keng Hua inequality, J. Math. Anal. Appl., 188 (1994), 700–702

  19. J. Pec̆arić, On Hua's inequality in real inner product spaces, Tamkang J. Math., 33 (2002), 265–268

  20. S. Radas and T. S̆ikić, A note on the generalization of Hua's inequality, Tamkang J. Math., 28 (1997), 321–323

  21. Takagi, H., Miura, T., Kanzo, T., Takahasi, S.E.: A reconsideration of Hua's inequality. J. Inequal. Appl. 1, 15–23 (2005)

    MathSciNet  MATH  Google Scholar 

  22. Wang, C.L.: Lo-Keng Hua inequality and dynamic programming. J. Math. Anal. Appl. 166, 345–350 (1992)

    Article  MathSciNet  Google Scholar 

  23. Wang, Y.Z., Huang, X.W.: Some inequalities about the positive elements in \(C^*\)-algebra. Chin. Q. J. Math. 2, 29–31 (1992)

    MATH  Google Scholar 

  24. Wegge-Olsen, N.E.: \(K\)-theory and \(C^*\)-algebra: a Friendly Approach. Oxford University Press (1993)

  25. Yang, G.S., Han, B.K.: A note on Hua's inequality for complex number. Tamkang J. Math. 27, 99–102 (1996)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgement

The authors thank the anonymous reviewers for valuable comments and suggestions that have improved the presentation of this paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    G. Q Hong & P. T. Li

Authors
  1. G. Q Hong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. T. Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. Q Hong.

Additional information

This research is supported by National Natural Science Foundation of China 11671201.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hong, G.Q., Li, P.T. On some new generalizations of Hua’s inequality. Acta Math. Hungar. 162, 62–75 (2020). https://doi.org/10.1007/s10474-020-01026-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10474-020-01026-5

Key words and phrases

Mathematics Subject Classification

Search

Navigation