Skip to main content
SpringerLink
Log in

Bounds of Some Divergence Measures Using Hermite Polynomial via Diamond Integrals on Time Scales

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

In this article, an inequality which contains bound of Csiszár divergence is generalised via diamond integral on time scales by utilizing the Hermite polynomial. Various constraints of Hermite polynomial are employed to provide some improvements of this new inequality. Bounds of different divergence measures are obtained by using particular convex functions. Furthermore, in seek of applications in mathematical statistics, bounds of different divergence measures are estimated on diverse fixed time scales. The paper addresses new results which are generalized (unified) form of both discrete and continuous results in literature (As time scales calculus unifies both discrete and continuous cases). Moreover, diamond integral can be used to study hybrid discrete-continuous systems.

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

Data Availibility

Data sharing is not applicable to this paper as no data sets were generated or analyzed during the current study.

References

  1. Adeel, M., Khan, K.A., Peĉarić, D., Peĉarić, J.: Estimation of \(f\)-divergence and Shannon entropy by using Levinson type inequalities for higher order convex functions via Hermite interpolating polynomial. J. Inequal. Appl. 2020 Article ID: 1 (2020)

  2. Adeel, M., Khan, K.A., Peĉarić, D., Peĉarić, J.: Estimation of f-divergence and Shannon entropy by Levinson type inequalities for higher order convex functions via Taylor polynomial. J. Math. Comput. Sci. 21(4), 322–334 (2020)

    Article  Google Scholar 

  3. Agarwal, R.P., Wong, P.J.Y.: Error Inequalities in Polynomial Interpolation and their Applications. Kluwer Academic Publishers, Dordrecht (1993)

    Book  Google Scholar 

  4. Ansari, I., Khan, K. A., Nosheen, A., Peĉarić, D., Peĉarić, J.: New entropic bounds on time scales via Hermite interpolating polynomial. J. Inequal. Appl. 2021, Article ID: 195 (2021)

  5. Beesack, P.: On the Green’s function of an \(N\)-point boundary value problem. Pac. J. Math. 12(3), 801–812 (1962)

    Article  MathSciNet  Google Scholar 

  6. Bibi, F., Bibi, R., Nosheen, A., Pečarić, J.: Extended Jensen’s functional for diamond integral via Green’s function and Hermite polynomial. J. Inequal. Appl. 2022, Article ID:50 (2022)

  7. Bilal, M., Khan, K.A., Nosheen, A., Pečarić, J.: Some inequalities related to Csiszár divergence via diamond integral on time scales. J. Inequal. Appl. 2023, Article ID:55 (2023)

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

    Book  Google Scholar 

  9. Bohner, M., Peterson, A.: Advances in Dynamic Equations on Time Scales. Birkhauser, Boston (2003)

    Book  Google Scholar 

  10. Brito, A.M.C., Martins, N., Torres, D.F.M.: The diamond integrals on time scales. Bull. Malays. Math. Sci. Soc. 38, 1453–1462 (2015)

    Article  MathSciNet  Google Scholar 

  11. Butt, S.I., Khan, K.A., Peĉarić, J.: Popoviciu type inequalities via Hermite’s polynomial. Math. Inequal. Appl. 19(4), 1309–1318 (2016)

    MathSciNet  Google Scholar 

  12. Butt, S.I., Khan, K.A., Peĉarić, J.: Popoviciu type inequalities via Green function and Hermite’s polynomial. Turk. J. Math. 40, 333–349 (2016)

    Article  Google Scholar 

  13. Chao, A., Jost, L., Hsieh, T.C., Ma, K.H., Sherwin, W.B., Rollins, L.A.: Expected Shannon entropy and Shannon differentiation between subpopulations for neutral genes under the finite island model. PLoS ONE 10(6), 1–24 (2015)

    Article  Google Scholar 

  14. Chow, C.K., Lin, C.N.: Approximating discrete probability distributions with dependence trees. IEEE Trans. Inf. Theory 14(3), 462–467 (1968)

    Article  Google Scholar 

  15. Čuljak, V., Arasgazić, G., Pečarić, J., Vukelić, A.: Generalization of Jensen’s inequality by Hermite polynomials and related results. In: Conference on inequalities and applications, 14: Book of Abstracts, p. 7

  16. Hilbbard, L.S.: Region segmentation using information divergence measures. Med. Image Anal. 8, 233–244 (2004)

    Article  Google Scholar 

  17. Horváth, L., Pečarić, D., Pečarić, J.: Estimations of f- and Renyi divergences by using a cyclic refinement of the Jensen’s inequality. Bull. Malays. Math. Sci. Soc. 42(3), 933–946 (2019)

    Article  MathSciNet  Google Scholar 

  18. Kilgore, T., Agarwal, R.P., Wong, P.J.Y.: Error inequalities in polynomial interpolation and their applications. J. Approx. Theory 86(3), 358–359 (1996)

    Article  Google Scholar 

  19. Levin, A.Y.: Some problems bearing on the oscillation of solution of linear differential equations. Sov. Math. Dokl. 4, 121–124 (1963)

    Google Scholar 

  20. Matić, M., Pearce, C.E.M., Pečarić, J.: Shannons and related inequalities in information theory. In: Survey on Classical Inequalities, pp. 127–164. Springer, Dordrecht (2000)

  21. Pardo, L.: Statistical Inference Based on Divergence measures. CRC Press, Boca Raton (2018)

    Book  Google Scholar 

  22. Peĉarić, J., Praljak, M.: Hermite interpolation and inequalities involving weighted averages of \(n\)-convex functions. Math. Inequal. Appl. 19(4), 69–80 (2016)

    MathSciNet  Google Scholar 

  23. Pečarić, J., Proschan, F., Tong, Y.L.: Convex Functions, Partial Orderings, and Statistical Applications. Academic Press, London (1992)

    Google Scholar 

  24. Sason, I., Verdú, S.: \(f\)-divergence inequalities. IEEE Trans. Inf. Theory 62, 5973–6006 (2016)

    Article  MathSciNet  Google Scholar 

  25. Sen, A.: On Economic Inequality. Oxford University Press, London (1973)

    Book  Google Scholar 

  26. Sheng, Q., Fadag, M., Henderson, J., Davis, J.M.: An exploration of combined dynamic derivatives on time scales and their applications. Nonlinear Anal. Real World Appl. 7(3), 395–413 (2006)

    Article  MathSciNet  Google Scholar 

  27. Smoljak Kalamir, K.: New diamond-\(\alpha \) Steffensen-type inequalities for convex functions over general time scale measure spaces. Axioms 2022(11), 323 (2022)

    Article  Google Scholar 

  28. Theil, H.: Economics and Information Theory. North-Holland, Amsterdam (1967)

    Google Scholar 

Download references

Acknowledgements

The authors wish to thanks the anonymous referees for their very careful reading of the manuscript and fruitful comments and suggestions.

Funding

There is no funding for this work.

Author information

Author notes

  1. Muhammad Bilal, Khuram Ali Khan, Ammara Nosheen and Josip Pečarić contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, University of Sargodha, Sargodha, Pakistan

    Muhammad Bilal, Khuram Ali Khan & Ammara Nosheen

  2. Government Associate College Miani, Sargodha, Pakistan

    Muhammad Bilal

  3. Croatian Academy of Science and Arts HR, Zagreb, Croatia

    Josip Pečarić

Authors
  1. Muhammad Bilal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Khuram Ali Khan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ammara Nosheen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Josip Pečarić
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

MB initiated the work and made calculations. KAK supervised and validated the draft. AN deduced the existing results and finalized the draft. JP dealt with the formal analysis and investigation. All the authors read and approved the final manuscript.

Corresponding author

Correspondence to Muhammad Bilal.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bilal, M., Khan, K.A., Nosheen, A. et al. Bounds of Some Divergence Measures Using Hermite Polynomial via Diamond Integrals on Time Scales. Qual. Theory Dyn. Syst. 23, 54 (2024). https://doi.org/10.1007/s12346-023-00911-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-023-00911-y

Keywords

Mathematics Subject Classification

Search

Navigation