New bounds for Shannon, Relative and Mandelbrot entropies via Hermite interpolating polynomial

• Autorzy: Mehmood N. , Butt S. I. , Pečarić D. , Pečarić J.

Identyfikatory

DOI

10.1515/dema-2018-0011

• Języki publikacji: EN

Abstrakty

EN

To procure inequalities for divergences between probability distributions, Jensen’s inequality is the key to success. Shannon, Relative and Zipf-Mandelbrot entropies have many applications in many applied sciences, such as, in information theory, biology and economics, etc. We consider discrete and continuous cyclic refinements of Jensen’s inequality and extend them from convex function to higher order convex function by means of different new Green functions by employing Hermite interpolating polynomial whose error term is approximated by Peano’s kernal. As an application of our obtained results, we give new bounds for Shannon, Relative and Zipf-Mandelbrot entropies.

Słowa kluczowe

EN

n-convex function; Hermite interpolating polynomial; new Green functions; Shannon entropy; relative entropy; Zipf-Mandelbrot entropy

PL

funkcja n-wypukła; interpolacja Hermite'a; funkcja Greena; entropia Shannona; entropia względna; entropia Zipfa-Mandelbrota

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2018
• Tom: Vol. 51, nr 1
• Strony: 112--130
• Opis fizyczny: Bibliogr. 37 poz.

Twórcy

autor

Mehmood N.

• Department of Mathematics, COMSATS, Institute of Information Technology, Lahore, Pakistan

autor

Butt S. I.

• Department of Mathematics, COMSATS, Institute of Information Technology, Lahore, Pakistan

autor

Pečarić D.

• Catholic University of Croatia, Ilica 242, Zagreb, Croatia

autor

Pečarić J.

• Faculty of Textile Technology, University of Zagreb, 10000 Zagreb, Croatia
• RUDN University, Miklukho-Maklaya str.6, 117198 Moscow, Russia

Bibliografia

• [1] Agarwal R. P., Wong P. J. Y., Error inequalities in polynomial interpolation and their applications, Kluwer Academic Publishers, Dordrecht, 1993
• [2] Beesack P. R., On the Green’s function of an N-point boundary value problem, Pacific J. Math., 1962, 12, 801-812
• [3] Levin A. Yu., Some problems bearing on the oscillation of solutions of linear differential equations, Soviet Math. Dokl., 1963, 4, 121-124
• [4] Widder D. V., Completely convex function and Lidstone series, Trans. Am. Math. Soc., 1942, 51, 387-398
• [5] Mehmood N., Agarwal R. P., Butt S. I., Pečarić J., New generalizations of Popoviciu-type inequalities via new Green’s functions and Montgomery identity, J. Inequal. Appl., 2017, 2017:108
• [6] Horváth L., Khan K. A., Pečarić J., Combinatorial improvements of Jensen’s inequality, Monographs in Inequalities, Element, Zagreb, 2014, 8
• [7] Horváth L., Khan K. A., Pečarić J., Cyclic refinements of the discrete and integral form of Jensen’s inequality with applications, Analysis, 2016, 36(4), 253-262, DOI: https://doi.org/10.1515/anly-2015-0022
• [8] Brnetić I., Khan K. A., Pečarić J., Refinement of Jensen’s inequality with applications to cyclic mixed symmetric means and Cauchy means, J. Math. Inequal., 2015, 9(4), 1309-1321
• [9] Horváth L., Inequalities corresponding to the classical Jensen’s inequality, J. Math. Inequal., 2009, 3(2), 189-200
• [10] Pečarić J., Proschan F., Tong Y. L., Convex functions, partial orderings and statistical applications, Academic Press, New York, 1992
• [11] Pečarić J., Praljak M., Witkowski A., Linear operator inequality for n−convex functions at a point, Math. Inequal. Appl., 2015, 18, 1201-1217
• [12] Csiszár I., Information measures: A critical survey, Trans. 7th Prague Conf. on Info. Th., Statist. Decis. Funct., Random Processes and 8th European Meeting of Statist, Volume B, Academia Prague, 1978, 73-86
• [13] Csiszár I., Information-type measures of diference of probability distributions and indirect observations, Studia Sci Math Hungar., 1967, 2, 299-318
• [14] Chao A., Jost L., Hsieh T. C., Ma K. H., Sherwin W. B., et al., Expected Shannon entropy and Shannon differentiation between subpopulations for neutral genes under the finite island model, PLOS ONE., 2015, 10(6), 1-24
• [15] Lesne A,. Shannon entropy: a rigorous notion at the crossroads between probability, information theory, dynamical systems and statistical physics, Mathematical Structures in Computer Science., 2014, 24(3), DOI:10.1017/S0960129512000783
• [16] Justice J. H. (Ed.), Maximum Entropy and Bayssian Methods in Applied Statistics, Cambridge University Press, Cambridge, 1986
• [17] Wędrowska E., Application of Kullback-Leibler relative entropy for studies on the divergence of household expenditures structures, Olsztyn Economic Journal., 2011, 6, 133-142
• [18] Kapur J. N., On the roles of maximum entropy and minimum discrimination information principles in Statistics, Technical Address of the 38th Annual Conference of the Indian Society of Agricultural Statistics., 1984, 1-44
• [19] Burbea I., Rao C. R., On the convexity of some divergence measures based on entropy functions, IEEE Transactions on Information Theory., 1982, 28, 489-495
• [20] Shannon C. E., A mathematical theory of communication, Bull. Sept. Tech. J., 1948, 27, 370-423 and 623-656
• [21] Frieden B. R., Image enhancement and restoration, In: T. S. Huang (Ed.), Picture Processing and Digital Filtering, Topics in Applied Physics, vol. 6, Springer, Berlin, Heidelberg, 1975
• [22] Leahy R. M., Goutis C. E., An optimal technique for constraint-based image restoration and mensuration, IEEE Trans. on Acoustics, Speech and Signal Processing., 1986, 34, 1692-1642
• [23] Kullback S., Information Theory and Statistics, J. Wiley, New York, 1959
• [24] Kullback S., Leibler R. A., On information and suflciency, Annals Math. Statist., 1951, 22, 79-86
• [25] Diodato V., Dictionary of Bibliometrics, Haworth Press, New York, 1994
• [26] Egghe L., Rousseau R., Introduction to Informetrics, Quantitative Methods in Library, Documentation and Information Science, Elsevier Science Publishers, New York, 1990
• [27] Piantadosi S. T., Zipf’s word frequency law in natural language: A critical review and future directions, Psychonomic Bulletin and Review, 2014, 21(5), 1112-1130
• [28] Montemurro M. A., Beyond the Zipf-Mandelbrot law in quantitative linguistics, (2001), arXiv:cond-mat/0104066v2 (22.2.2017)
• [29] Manin D., Mandelbrot’s model for Zipf’s law: Can Mandelbrot’s model explain Zipf’s law for language, Journal of Quantitative Linguistics, 2009, 16(3), 274-285
• [30] Silagadze Z. K., Citations and the Zipf–Mandelbrot law, Complex Systems, 1997, 11, 487-499
• [31] Mouillot D, Lepretre A., Introduction of relative abundance distribution (RAD) indices, estimated from the rank-frequency diagrams (RFD), to assess changes in community diversity, Environmental Monitoring and Assessment, Springer, 2000, 63(2), 279-295
• [32] Butt S. I., Pečarić J., Popoviciu’s inequality for n-convex functions, Lap Lambert Academic Publishing, ISBN: 978-3-659-81905-6 2016
• [33] Cerone P., Dragomir S. S., Some new Ostrowski-type bounds for the Čebyšev functional and applications, J. Math. Inequal., 2014, 8(1), 159-170
• [34] Jakšetic J., Pečarić J., Exponential convexity method, J. Convex. Anal., 2013, 20(1), 181–197
• [35] Pečarić J., Perić J., Improvement of the Giaccardi and the Petrović inequality and related Stolarsky type means, An. Univ. Craiova Ser. Mat. Inform., 2012, 39(1), 65-75
• [36] Butt S. I., Pečarić J., Vukelić A., Generalization of Popoviciu type inequalities Via Fink’s identity, Mediterr. J. Math., 2016, 13(4), 1495-1511
• [37] Butt S. I., Khan K. A., Pečarić J., Popoviciu type inequalities via Green function and generalized Montgomery identity, Math. Inequal. Appl., 2015, 18(4), 1519-1538

Uwagi

PL

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).