Rad HAZU, Matematičke znanosti, Vol. 24 (2020), 59-80.

COMBINATORIAL EXTENSIONS OF POPOVICIU'S INEQUALITY VIA ABEL-GONTSCHAROFF POLYNOMIAL WITH APPLICATIONS IN INFORMATION THEORY

Saad Ihsan Butt, Tahir Rasheed, Đilda Pečarić and Josip Pečarić

Catholic University of Croatia, Ilica 242, Zagreb, Croatia
e-mail: gildapeca@gmail.com

RUDN University, Miklukho-Maklaya str.6, 117198 Moscow, Russia
e-mail: pecaric@element.hr

Abstract.   We establish new refinements and improvements of Popoviciu's inequality for n-convex functions using Abel-Gontscharoff interpolating polynomial along with the aid of new Green functions. We construct new inequalities for n-convex functions and compute new upper bounds for Ostrowski and Grüss type inequalities. As an application of our work in information theory, we give new estimations for Shannon, Relative and Zipf-Mandelbrot entropies using generalized Popoviciu's inequality.

2020 Mathematics Subject Classification.   26A51, 26D15, 26E60, 94A17, 94A15.

Key words and phrases.   Popoviciu's inequality, n-convex function, new Green functions, Grüss and Ostrowski inequality, divergence functional, Shannon Entropy, Kullback-Liebler distance.

DOI: https://doi.org/10.21857/mzvkptqwg9

References:

1. R. P. Agarwal and P. J. Y. Wong, Error Inequalities in Polynomial Interpolation and their Applications, Kluwer Academic Publishers, Dordrecht, 1993.
   MathSciNet     CrossRef

2. M. Bencze, C. P. Niculescu and F. Popovici, Popoviciu' s inequality for functions of several variables, J. Math. Anal. Appl. 365 (2010), 399-409.
   MathSciNet     CrossRef

3. S. I. Butt, K. A. Khan and J. Pečarić, Popoviciu type inequalities via Green function and generalized Montgomery identity, Math. Inequal. Appl. 18 (2015), 1519-1538.
   MathSciNet     CrossRef

4. S. I. Butt, K. A. Khan and J. Pečarić, Popoviciu type inequalities via Green function and Taylor polynomial, Turkish J. Math. 40 (2016), 333-349.
   MathSciNet     CrossRef

5. S. I. Butt, J. Pečarić and A. Vukelić, Generalization of Popoviciu-type inequalities via Fink's identity, Mediterr. J. Math. 13 (2016), 1495-1511.
   MathSciNet     CrossRef

6. P. Cerone and S. S. Dragomir, Some new Ostrowski-type bounds for the Čebyšev functional and applications, J. Math. Inequal. 8 (2014), 159-170.
   MathSciNet     CrossRef

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

8. V. Diodato, Dictionary of Bibliometrics, Haworth Press, New York, 1994.

9. L. Egghe and R. Rousseau, Introduction to Informetrics, Quantitative Methods in Library, Documentation and Information Science, Elsevier Science Publishers, New York, 1990.

10. L. Horváth, Đ. Pečarić and J. Pečarić, Estimations of f- and Rényi divergences by using a cyclic refinement of the Jensen's inequality, Bull. Malays. Math. Sci. Soc. 42 (2019), 933-946.
    MathSciNet     CrossRef

11. J. Jakšetić, Đ. Pečarić and J. Pečarić, Some properties of Zipf-Mandelbrot law and Hurwitz ζ-function, Math. Inequal. Appl. 21 (2018), 575-584.
    MathSciNet     CrossRef

12. J. Jakšetić and J. Pečarić, Exponential convexity method, J. Convex. Anal. 20 (2013), 181-197.
    MathSciNet

13. M. A. Khan, Đ. Pečarić and J. Pečarić, Bounds for Shannon and Zipf-Mandelbrot entropies, Math. Methods Appl. Sci. 40 (2017), 7316-7322.
    MathSciNet     CrossRef

14. M. A. Khan, Đ. Pečarić and J. Pečarić, On Zipf-Mandelbrot entropy, J. Comput. Appl. Math. 346 (2019) 192-204.
    MathSciNet     CrossRef

15. S. Kullback, Information Theory and Statistics, J. Wiley, New York, 1959.
    MathSciNet

16. A. Lesne, Shannon entropy: a rigorous notion at the crossroads between probability, information theory, dynamical systems and statistical physics, Math. Structures Comput. Sci. 24 (2014), no. 3, e240311, 63 pp.
    MathSciNet     CrossRef

17. D. Manin, Mandelbrot's model for Zipf's Law: Can Mandelbrot's model explain Zipf's law for language, Journal of Quantitative Linguistics 16(3) (2009), 274-285.
    CrossRef

18. N. Mehmood, R. P. Agarwal, S. I. Butt and J. Pečarić, New generalizations of Popoviciu-type inequalities via new Green's functions and Montgomery identity, J. Inequal. Appl. 2017 (2017), Paper No. 108.
    MathSciNet     CrossRef

19. M. V. Mihai and F. C. Mitroi-Symeonidis, New extensions of Popoviciu's inequality, Mediterr. J. Math. 13 (2016) 3121-3133.
    MathSciNet     CrossRef

20. M. A. Montemurro, Beyond the Zipf-Mandelbrot law in quantitative linguistics, Physica A: Statistical Mechanics and its Applications 300(3–4) (2001), 567-578.
    CrossRef

21. C. P. Niculescu, The Integral version of Popoviciu's inequality, J. Math. Inequal. 3 (2009), 323-328.
    MathSciNet     CrossRef

22. C. P. Niculescu and F. Popovici, A refinement of Popoviciu's inequality, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 49 (2006), 285-290.
    MathSciNet

23. J. Pečarić and J. Perić, Improvement of the Giaccardi and the Petrović inequality and related Stolarsky type means, An. Univ. Craiova Ser. Mat. Inform. 39 (2012), 65-75.
    MathSciNet

24. J. Pečarić, M. Praljak and A. Witkowski, Linear operator inequality for n-convex functions at a point, Math. Inequal. Appl. 18 (2015), 1201-1217.
    MathSciNet     CrossRef

25. J. Pečarić, F. Proschan and Y. L. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, Boston, 1992.
    MathSciNet

26. S. T. Piantadosi, Zipf's word frequency law in natural language: A critical review and future directions, Psychonomic Bulletin and Review 21(5) (2014), 1112-1130.
    CrossRef

27. T. Popoviciu, Sur certaines inégalités qui caractérisent les fonctions convexes, An. Sti. Univ. "Al. I. Cuza" Iasi, Sect. I a Mat. (N.S.) 11B (1965), 155-164.
    MathSciNet