Skip to main content
SpringerLink
Log in

Estimation of different entropies via Hermite interpolating polynomial using Jensen type functionals

  • Original Research Paper
  • Published:
The Journal of Analysis Aims and scope Submit manuscript

Abstract

We estimated the different entropies like Shannon entropy, Rényi divergences, Csiszar divergence by using the Jensen’s type functionals. The Zipf’s mandelbrot law and hybrid Zipf’s mandelbrot law are used to estimate the Shannon entropy. Further the Hermite interpolating polynomial is used to generalize the new inequalities for m-convex 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

References

  1. Agarwal, R.P., and P.J.Y. Wong. 1993. Error inequalities in polynomial interpolation and their applications. Dordrecht: Kluwer Academic Publishers.

    Book  Google Scholar 

  2. Agarwal, P., S.S. Dragomir, M. Jleli, and B. Samet (eds.). 2018. Advances in mathematical inequalities and applications. Basel: Birkhäuser.

    MATH  Google Scholar 

  3. Agarwal, P., M. Jleli, and M. Tomar. 2017. Certain Hermite-Hadamard type inequalities via generalized k-fractional integrals. Journal of Inequalities and Applications 2017 (1): 55.

    Article  MathSciNet  Google Scholar 

  4. Anderson, G., and Y. Ge. 2005. The size distribution of Chinese cities. Regional Science and Urban Economics 35(6): 756–776.

    Article  Google Scholar 

  5. Auerbach, F. 1913. Das Gesetz der Bevölkerungskonzentration. Petermanns Geographische Mitteilungen 59: 74–76.

    Google Scholar 

  6. Beesack, P. 1962. On the Green’s function of an \(N\)-point boundary value problem. Pacific Journal of Mathematics 12(3): 801–812.

    Article  MathSciNet  Google Scholar 

  7. Black, D., and V. Henderson. 2003. Urban evolution in the USA. Journal of Economic Geography 3(4): 343–372.

    Article  Google Scholar 

  8. Bosker, M., S. Brakman, H. Garretsen, and M. Schramm. 2008. A century of shocks: The evolution of the German city size distribution 1925–1999. Regional Science and Urban Economics 38(4): 330–347.

    Article  Google Scholar 

  9. Butt, S.I., K.A. Khan, and J. Pečarić. 2015. Generaliztion of Popoviciu inequality for higher order convex function via Tayor’s polynomial. Acta Univ. Apulensis Math. Inform 42: 181–200.

    MATH  Google Scholar 

  10. Butt, S.I., N. Mehmood, and J. Pečarić. 2017. New generalizations of Popoviciu type inequalities via new green functions and Fink’s identity. Transactions of A. Razmadze Mathematical Institute 171(3): 293–303.

    Article  MathSciNet  Google Scholar 

  11. Butt, S.I., and J. Pečarić. 2016. Popoviciu’s inequality For \(N\)-convex functions. Saarbrücken: Lap Lambert Academic Publishing.

  12. Butt, S. I., and J. Pečarić, 2015. Weighted Popoviciu type inequalities via generalized Montgomery identities. Hrvatske akademije znanosti i umjetnosti: Matematicke znanosti, (523= 19), 69-89.

  13. Butt, S.I., K.A. Khan, and J. Pečarić. 2016. Popoviciu type inequalities via Hermite’s polynomial. Mathematical Inequalities & Applications 19 (4): 1309–1318.

    Article  MathSciNet  Google Scholar 

  14. Csiszár, I. 1978. Information measures: A critical survey. In: Tans. 7th Prague Conf. on Info. Th., Statist. Decis. Funct., Random Process and 8th European Meeting of Statist., vol. B, pp. 73–86, Academia Prague.

  15. Csiszár, I. 1967. Information-type measures of difference of probability distributions and indirect observations. Stud. Sci. Math. Hungar. 2: 299–318.

    MathSciNet  MATH  Google Scholar 

  16. Horváth, L. 2011. A method to refine the discrete Jensen’s inequality for convex and mid-convex functions. Mathematical and Computer Modelling 54(9–10): 2451–2459.

    Article  MathSciNet  Google Scholar 

  17. Horváth, L., K.A. Khan, and J. Pečarić. 2014. Combinatorial improvements of Jensens inequality/classical and new refinements of Jensens inequality with applications, monographs in inequalities 8. Zagreb: Element.

    Google Scholar 

  18. Horváth, L., Khan, K.A., and J. Pečarić. 2014. Refinement of Jensen’s inequality for operator convex functions. Adv. Inequal. Appl., 2014: 26

    Article  Google Scholar 

  19. Horváth, L., and J. Pečarić. 2011. A refinement of discrete Jensen’s inequality. Math. Inequal. Appl. 14: 777–791.

    Article  MathSciNet  Google Scholar 

  20. Horváth, L., Đ. Pečarić, and J. Pečarić. 2017. Estimations of f-and Rényi divergences by using a cyclic refinement of the Jensen’s inequality. Bulletin of the Malaysian Mathematical Sciences Society 1–14.

  21. Ioannides, Y.M., and H.G. Overman. 2003. Zipf’s law for cities: An empirical examination. Regional Science and Urban Economics 33(2): 127–137.

    Article  Google Scholar 

  22. Khan, K. A., Niaz, T., Pečarić, D., and J. Pečarić. 2018. Refinement of Jensen’s Inequality and Estimation of f- and Renyi Divergence via Montgomery identity. J. Inequal. Appl. 2018(1): 318.

    Article  Google Scholar 

  23. Kullback, S. 1997. Information theory and statistics. New York: Dover Publications.

    MATH  Google Scholar 

  24. Kullback, S., and R.A. Leibler. 1951. On information and sufficiency. The Annals of Mathematical Statistics 22(1): 79–86.

    Article  MathSciNet  Google Scholar 

  25. Levin, A.Y. 1963. Some problems bearing on the oscillation of solution of liear differential equations. Soviet Math. Dokl. 4: 121–124.

    Google Scholar 

  26. Lovričevic, N., Đ. Pečarić, and J. Pečarić. 2018. Zipf-Mandelbrot law, f-divergences and the Jensen-type interpolating inequalities. Journal of Inequalities and Applications 2018(1): 36.

    Article  MathSciNet  Google Scholar 

  27. Matic, M., Pearce, C.E., and J. Pečarić, 2000. Shannon’s and related inequalities in information theory. In Survey on Classical Inequalities (pp. 127-164). Springer, Dordrecht.

  28. Mehrez, K., and P. Agarwal. 2019. New Hermite-Hadamard type integral inequalities for convex functions and their applications. Journal of Computational and Applied Mathematics 350: 274–285.

    Article  MathSciNet  Google Scholar 

  29. Niaz, T., K.A. Khan, and J. Pečarić. 2017. On generalization of refinement of Jensen’s inequality using Fink’s identity and Abel-Gontscharoff Green function. Journal of Inequalities and Applications 2017(1): 254.

    Article  MathSciNet  Google Scholar 

  30. Pečarić, J., Khan, K. A., and I. Perić, 2014. Generalization of Popoviciu type inequalities for symmetric means generated by convex functions, J. Math. Comput. Sci., Vol. 4, No. 6, 1091-1113.

  31. Pečarić, J., F. Proschan, and Y.L. Tong. 1992. Convex functions. Partial Orderings and Statistical Applications: Academic Press, New York.

  32. Rényi, A. 1960. On measure of information and entropy. In: Proceeding of the Fourth Berkely Symposium on Mathematics, Statistics and Probability, pp. 547-561.

  33. Rosen, K.T., and M. Resnick. 1980. The size distribution of cities: An examination of the Pareto law and primacy. Journal of Urban Economics 8(2): 165–186.

    Article  Google Scholar 

  34. Ruzhansky, M., Cho, Y. J., Agarwal, P., and I. Area, (Eds.). 2017. Advances in real and complex analysis with applications. Springer Singapore.

  35. Soo, K.T. 2005. Zipf’s Law for cities: A cross-country investigation. Regional science and urban Economics 35(3): 239–263.

    Article  Google Scholar 

  36. Tomar, M., P. Agarwal, and J. Choi. 2020. Hermite-Hadamard type inequalities for generalized convex functions on fractal sets style. Boletim da Sociedade Paranaense de Matemática 38(1): 101–116.

    Article  MathSciNet  Google Scholar 

  37. Widder, D.V. 1942. Completely convex function and Lidstone series. Trans. Am. Math. Soc. 51: 387–398.

    Article  MathSciNet  Google Scholar 

  38. Zipf, G.K. 1949. Human behaviour and the principle of least-effort. Cambridge, MA ed. Reading: Addison-Wesley.

Download references

Acknowledgements

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

Funding

The first three authors has no funding support for this research work. The research of 4th author was supported by the Ministry of Education and Science of the Russian Federation (the Agreement number No. 02.a03.21.0008).

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Sargodha, Sargodha, 40100, Pakistan

    Khuram Ali Khan & Tasadduq Niaz

  2. Department of Mathematics, University of Central Punjab, Lahore, Pakistan

    Tasadduq Niaz

  3. Catholic University of Croatia, Ilica 242, Zagreb, Croatia

    Đilda Pečarić

  4. RUDN University, Miklukho-Maklaya str. 6, Moscow, Russia, 117198

    Josip Pečarić

Authors
  1. Khuram Ali Khan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tasadduq Niaz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Đilda Pečarić
    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

All authors jointly worked on the results and they read and approved the final manuscript.

Corresponding author

Correspondence to Tasadduq Niaz.

Ethics declarations

Conflict of interest

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

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

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

Khan, K.A., Niaz, T., Pečarić, Đ. et al. Estimation of different entropies via Hermite interpolating polynomial using Jensen type functionals. J Anal 29, 15–46 (2021). https://doi.org/10.1007/s41478-020-00245-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41478-020-00245-x

Keywords

Mathematics Subject Classification

Search

Navigation