Skip to main content

Advertisement

SpringerLink
Log in

Some new dynamic Hardy-type inequalities with kernels involving monotone functions

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

Abstract

The main objective of the present paper is an extension of several classical integral Hardy-type inequalities with kernels to a time scales setting. The established inequalities refer to a class of monotone functions and they are characterized by appropriate relations for the accompanying weight functions. As an application, we obtain the corresponding discrete Hardy-type inequalities for monotone sequences, which are essentially new.

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.

Similar content being viewed by others

References

  1. Agarwal, R.P., O’Regan, D., Saker, S.H.: Dynamic Inequalities on Time Scales. Springer, Berlin (2014)

    Book  Google Scholar 

  2. Agarwal, R.P., O’Regan, D., Saker, S.H.: Hardy Type Inequalities on Time Scales. Springer, Berlin (2016)

    Book  Google Scholar 

  3. Bennett, G., Grosse-Erdmann, K.-G.: Weighted Hardy inequalities for decreasing sequences and functions. Math. Ann. 334, 489–531 (2006)

    Article  MathSciNet  Google Scholar 

  4. Bibi, R., Bohner, M., Pečarić, J., Varošanec, S.: Minkowski and Beckenbach–Dresher inequalities and functionals on time scales. J. Math. Inequal. 7, 299–312 (2013)

    Article  MathSciNet  Google Scholar 

  5. Bohner, M., Georgiev, S.G.: Multiple Integration on Time Scales, Multivariable Dynamic Calculus on Time Scales, pp. 449–515. Springer, Cham (2016)

    Book  Google Scholar 

  6. Bohner, M., Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston (2001)

    Book  Google Scholar 

  7. Bohner, M., Peterson, A.: Advances in Dynamic Equations on Time Scales. Springer Science and Business Media, New York (2002)

    MATH  Google Scholar 

  8. Hardy, G.H.: Notes on some points in the integral calculus, LX. An inequality between integrals. Messenger Math. 54, 150–156 (1925)

    Google Scholar 

  9. Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1967)

    MATH  Google Scholar 

  10. Heinig, H., Maligranda, L.: Weighted inequalities for monotone and concave functions. Stud. Math. 116, 133–165 (1995)

    MathSciNet  MATH  Google Scholar 

  11. Kufner, A., Maligranda, L., Persson, L.-E.: The Hardy inequality—about its history and some related results. Vydavatelský Servis Publishing House, Pilsen (2007)

    MATH  Google Scholar 

  12. Lai, S.: Weighted norm inequalities for general operators on monotone functions. Trans. Am. Math. Soc. 340, 811–836 (1993)

    Article  MathSciNet  Google Scholar 

  13. Magyar, A., Stein, E.M., Wainger, S.: Discrete analogues in harmonic analysis: spherical averages. Ann. Math. 155, 189–208 (2002)

    Article  MathSciNet  Google Scholar 

  14. Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165, 207–226 (1972)

    Article  MathSciNet  Google Scholar 

  15. Muckenhoupt, B.: Hermite conjugate expansions. Trans. Am. Math. Soc. 139, 243–260 (1969)

    Article  MathSciNet  Google Scholar 

  16. Muckenhoupt, B.: Mean convergence of Hermite and Laguerre series II. Trans. Am. Math. Soc. 147, 433–460 (1970)

    Article  MathSciNet  Google Scholar 

  17. Muckenhoupt, B., Stein, E.M.: Classical expansions and their relation to conjugate harmonic functions. Trans. Am. Math. Soc. 118, 17–92 (1965)

    Article  MathSciNet  Google Scholar 

  18. Oguntuase, J.A., Persson, L.E.: Time scales Hardy-type inequalities via superquadracity. Ann. Funct. Anal. 5, 61–73 (2014)

    Article  MathSciNet  Google Scholar 

  19. Opic, B., Kufner, A.: Hardy-Type Inequalities, Pitman Research Notes in Mathematics Series. Longman Scientific and Technical, Harlow (1990)

    MATH  Google Scholar 

  20. Řehak, P.: Hardy inequality on time scales and its application to half-linear dynamic equations. J. Inequal. Appl. 2005(5), 495–507 (2005)

    Article  MathSciNet  Google Scholar 

  21. Saker, S.H., Awwad, E., Saied, A.: Some new dynamic inequalities involving monotonic functions on time scales. J. Funct. Spaces, Article ID 7584836 (2019)

  22. Saker, S.H., Mahmoud, R.R., Peterson, A.: Weighted Hardy-type inequalities on time scales with applications. Mediterr. J. Math. 13, 585–606 (2016)

    Article  MathSciNet  Google Scholar 

  23. Saker, S.H., O’Regan, D.: Hardy and Littlewood inequalities on time scales. Bull. Malays. Math. Sci. Soc. 39, 527–543 (2016)

    Article  MathSciNet  Google Scholar 

  24. Saker, S.H., Rezk, H.M., Krnić, M.: More accurate dynamic Hardy-type inequalities obtained via superquadraticity. RACSAM 113, 2691–2713 (2019)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt

    Samir H. Saker

  2. Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt

    Ahmed I. Saied

  3. Faculty of Electrical Engineering and Computing, University of Zagreb, Unska 3, 10000, Zagreb, Croatia

    Mario Krnić

Authors
  1. Samir H. Saker
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ahmed I. Saied
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mario Krnić
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mario Krnić.

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

Saker, S.H., Saied, A.I. & Krnić, M. Some new dynamic Hardy-type inequalities with kernels involving monotone functions. RACSAM 114, 142 (2020). https://doi.org/10.1007/s13398-020-00876-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13398-020-00876-6

Keywords

Mathematics Subject Classification

Search

Navigation