Estimation of Shannon entropy and divergences by Levinson-type inequalities using new Green functions via Lidstone polynomial

Abstract

The paper is devoted to obtain generalized results related to Levinson type inequalities for $(2n+1)$-convex functions $(n>1)$ using new Green functions, along with Lidstone interpolating polynomial. Some new estimations of bounds for Csiszár divergence and Shannon entropies are derived using novel functionals. Furthermore, various new inequalities involving Kullback-Leibler divergence, Bhattacharyya coefficient, Jeffrey's distance and Triangular discrimination are presented.

Keywords

• Levinson’s inequality
• Jeffrey’s distance
• Lidstone polynomial
• Shannon entropy

References

1. [1] M. Adeel, K. A. Khan, Ð. Pečarić and J. Pečarić, Generalization of the Levinson inequality with applications to information theory, Journal of Inequalities and Applications, 2019(1), 230, 2019.
2. [2] M. Adeel, K. A. Khan, Ð. Pečarić and J. Pečarić, Levinson type inequalities for higher order convex functions via Abel-Gontscharoff interpolation, Advances in Difference equations, 2019(1), 430, 2019.
3. [3] M. Adeel, K. A. Khan, Ð. Pečarić and J. Pečarić, Estimation of f-divergence and Shannon entropy by Levinson type inequalities via new Green’s functions and Lidstone polynomial, Advances in Difference equations, 2020(1), 27, 2020.
4. [4] M. Adeel, K. A. Khan, Ð. Pečarić and J. Pečarić, Estimation of f-divergence and Shannon entropy by using Levinson type Inequalities for higher order convex functions via Hermite interpolating polynomial, Journal of Inequalities and Applications, 2020(1), 137, 2020.
5. [5] M. Adeel, K. A. Khan, Ð. Pečarić and J. Pečarić, Levinson-type inequalities via new Green functions and Montgomery identity, Open Mathematics, 18(1), 632-652, 2020.
6. [6] M. Adeel, K. A. Khan, Ð. Pečarić and J. Pečarić, Estimation of f-divergence and Shannon entropy by Levinson type inequalities via Lidstone interpolating polynomial, Transactions of A. Razmadze Mathematical Institute, 175(1), 1-11, 2021.
7. [7] M. Adeel, K. A. Khan, Ð. Pečarić and J. Pečarić, Estimation of f-divergence and Shannon entropy by Bullen type inequalities via Finks identity, Filomat, 36(2), 527- 538, 2022.
8. [8] M. Adeel, K. A. Khan, Ð. Pečarić and J. Pečarić, Entropy results for Levinson-type inequalities via Green functions and Hermite interpolating polynomial, Aequationes mathematicae, 96(1), 1-16, 2022.

9. [9] R. P. Agarwal and P. J. Y. Wong, Error inequalities in polynomial interpolation and their applications, Kluwer Academic Publishers, Dordrecht, 1993.
10. [10] M. Bilal, K. A. Khan, A. Nosheen, and J. Pečarić, Generalizations of Shannon type inequalities via diamond integrals on time scales, Journal of Inequalities and Applications, 2023(1), 24, 2023.
11. [11] M. Bilal, K. A. Khan, A. Nosheen, and J. Pečarić, Some inequalities related to Csiszár divergence via diamond integral on time scales, Journal of Inequalities and Applications, 2023(1), 55, 2023.
12. [12] M. Bilal, K. A. Khan, A. Nosheen and J. Pečarić, Bounds of Some Divergence Measures Using Hermite Polynomial via Diamond Integrals on Time Scales, Qualitative Theory of Dynamical Systems, 23(2), 54, 2024.
13. [13] P.S. Bullen, An inequality of N. Levinson, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 412/460, 109-112, 1973.
14. [14] S. I. Butt, K. A. Khan and J. Peˇcari´c, Generalization of Popoviciu inequality for higher order convex function via Taylors polynomial, Acta Universitatis Apulensis, Mathematics-Informatics, 42, 181-200, 2015.
15. [15] S. I. Butt, K. A. Khan and J. Peˇcari´c, Popoviciu type inequalities via Hermites polynomial, Mathematical Inequalities and Applications, 19(4), 1309-1318, 2016.
16. [16] S. I. Butt, N. Mehmood, and J. Peˇcari´c, New generalizations of Popoviciu type inequalities via new Green functions and Fink’s identity, Transactions of A.Razmadze Mathematical Institute, 171(3), 293-303, 2017.
17. [17] S. I. Butt and J. Peˇcari´c, Weighted Popoviciu type inequalities via generalized Montgomery identities, Rad HAZU, Matemati˘cke Znanosti, 19(523), 69-89, 2015.
18. [18] S. I. Butt and J. Peˇcari´c, Popoviciu’s Inequality for N-Convex Functions, Lap Lambert Academic Publishing, 2016.
19. [19] C. K. C. N. Chow, and C. Liu, Approximating discrete probability distributions with dependence trees, IEEE transactions on Information Theory, 14(3), 462-467, 1968.
20. [20] I. Csiszár, Information-type measures of difference of probability distributions and indirect observations, Studia Scientiarum Mathematicarum Hungarica, 2, 299-318, 1967.
21. [21] I. Csiszár, Information measures: a critical survey, In: Tans. 7th Prague Conf. on Info. Th., Statist. Decis. Funct., Random Process and 8th European Meeting of Statist., B, 73-86, Academia Prague, 1978.
22. [22] L. Horváth, Ð. Pečarić and J. Pečarić, Estimations of f-and Rényi divergences by using a cyclic refinement of the Jensens inequality, Bulletin of the Malaysian Mathematical Sciences Society, 42, 933-946, 2019.
23. [23] M. A. Khan, Ð. Peˇcari´c and J. Peˇcari´c, A new refinement of the Jensen inequality with applications in information theory, Bulletin of the Malaysian Mathematical Sciences Society, 44(1), 267-278, 2021.
24. [24] N. Levinson, Generalization of an inequality of Kay Fan, Journal of Mathematical Analysis and Applications, 6, 133-134, 1969.
25. [25] N. Mehmood, R. P. Agarwal, S. I. Butt and J. Peˇcari´c, New generalizations of Popoviciu type inequalities via new Greens functions and Montgomery identity, Journal of Inequalities and Applications, 2017, 121, 2017.
26. [26] A. M. Mercer, 94.33 Short proofs of Jensen’s and Levinson’s Inequalities, The Mathematical Gazette, 94(531), 492-495, 2010.
27. [27] J. Pečarić, On an inequality on N. Levinson, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 412/460 109-112, 1980.
28. [28] J. Pečarić, F.Proschan and Y. L. Tong, Convex functions, Partial Orderings and Statistical Applications, Academic Press, New York, 1992.
29. [29] T. Popoviciu, Sur une inegalite de N, Levinson Mathematica (Cluj), 6, 301-306, 1969.
30. [30] A. Rasheed, K. A. Khan, J. Pečarić and Ð. Pečarić, Generalization of the Levinson inequality via new Green functions with applications to information theory, Journal of Inequalities and Applications, 2023(1), 124, 2023.
31. [31] A. Rasheed, K. A. Khan, J. Pečarić and Ð. Pečarić, Generalizations of Levinson-type inequalities via new Green functions and Hermite interpolating polynomial, Journal of inequalities and applications, 2024(1), 70, 2024.
32. [32] A. Sen, M. A. Sen, J. E. Foster, S. Amartya and J. E. Foster, On economic inequality, Oxford university press, 1997.
33. [33] C. E. Shannon, A mathematical theory of communication, The Bell system technical journal, 27(3), 379-423, 1948.
34. [34] H. Theil, Economics and Information Theory, North-Holland, Amsterdam, 1967.
35. [35] D. V. Widder, Completely convex function and Lidstone series, Trans. Am. Math. Soc. 51, 387-398, 1942.