Document Type : Original Article

Authors

1 University of Rijeka, Faculty of Civil Engineering, Radmile Matejcic 3, 51 000 Rijeka, Croatia

2 Catholic University of Croatia, Ilica 242, 10 000 Zagreb, Croatia

3 Croatian Academy of Sciences and Arts, 10 000 Zagreb, Croatia

10.30495/maca.2024.2017279.1093

Abstract

In this paper, we study 3-convex functions, which are characterized by the third-order divided differences, and for them, we derive a class of inequalities of the Jensen and Edmundson-Lah-Ribarič type involving positive linear functionals that do not require convexity in the classical sense. A great number of theoretic divergences, i.e. measures of distance between two probability distributions, are special cases of Csiszár f-divergence for different choices of the generating function f. In the second part of this paper, we apply our main results to the generalized f-divergence functional to obtain lower and upper bounds. Examples of Zipf–Mandelbrot law are used to illustrate the results. In addition, the obtained results are utilized in constructing some families of exponentially convex functions and Stolarsky-type means.

Keywords

[1] S. Abramovich, Quasi-arithmetic means and subquadracity, J. Math. Inequal., 9(4) (2015), 1157–1168.
[2] P. R. Beesack and J. E. Pecaric, On the Jessen’s inequality for convex functions, J. Math. Anal. 110(1985), 536–552.
[3] M. Ben Bassat, f-entropies, probability of error, and feature selection, Inf. Contr., 39 (1978), 227–242.
[4] P. S. Bullen, D. S. Mitrinovic, and P. M. Vasic,  Means and their inequalities, D. Reidel Publishing Co., Dordrecht, Boston, Lancaster and Tokyo, 1987.
[5] C. H. Chen, Statistical pattern recognition, Rochelle Park, NJ: Hayden Book Co., 1973.
[6] C. K. Chow and C. N. Liu, Approximating discrete probability distributions with dependence trees, IEEE Trans. Inf. Theory, 14(3) (1968), 462–467.
[7] I. Csiszar, 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, pp. 73–86.
[8] I. Csiszar, Information-type measures of difference of probability functions and indirect observations, Studia Sci. Math. Hungar., 2 (1967), 299–318.
[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. Fry Richardson, Statistics of deadly quarrels, Marcel Dekker, 1960.
[11] D. V. Gokhale and S. Kullback, Information in contingency tables, Pacific Grove, Boxwood Press 1978.
[12] L. Horvath, Weighted form of a recent refinement of the discrete Jensen’s inequality, Math. Inequal. Appl., 17(3), (2014), 947–961.
[13] L. Horvath and J. Pecaric, A refinement of the discrete Jensen’s inequality, Math. Inequal. Appl., 14(4) (2011), 777–791.
[14] E. Isaacson and H. B. Keller, Analysis of numerical methods, Dover Publications Inc., New York, 1966.
[15] S. Ivelic and J. Pecaric, Generalizations of converse Jensen’s inequality and related results, J. Math. Inequal., 5(1) (2011), 43–60.
[16] J. Jaksetic and J. Pecaric, Exponential convexity method, J. Conv. Anal., 20(1) (2013), 181–197.
[17] R. Jaksic and J. Pecaric, New converses of the Jessen and Lah-Ribariˇc inequalities II, J. Math. Inequal., 7(4) (2013), 617–645.
[18] R. Jaksic and J. Pecaric, Levinson’s type generalization of the Edmundson-Lah-Ribaric inequality, Mediterr. J. Math., 13(1) (2016), 483–496.
[19] B. Jessen, Bemaerkinger om konvekse funktioner og uligheder imellem middelvaerdier I, Mat. Tidsskrift, B (1931), 17–28.
[20] T. Kailath, The divergence and Bhattacharyya distance measures in signal selection, IEEE Trans. Commun. Technol., 15(1) (1967), 52–60.
[21] M. Krnic, R. Mikic, and J. Pecaric, Strengthened converses of the Jensen and Edmundson-Lah-Ribariˇc inequalities, Adv. Oper. Theory, 1(1) (2016), 104–122.
[22] K. Krulic Himmelreich, J. Pecaric, D. Pokaz, Inequalities of Hardy and Jensen / New Hardy type inequalities with general kernels, Monographs in inequalities 6, Element, Zagreb, 2013.
[23] J. Liang and G. Shi, Comparison of differences among power means Qr,α(a, b, x)s, J. Math. Inequal., 9(2) (2015), 351–360.
[24] J. Lin and S. K. M. Wong, Approximation of discrete probability distributions based on a new divergence measure, Congur. Numeran., 61 (1988), 75–80.
[25] B. Manaris, D. Vaughan, C. S. Wagner, J. Romero, and R. B. Davis, Evolutionary music and the Zipf-Mandelbrot law: Developing fitness functions for pleasant music, Proc. 1st Eur. Workshop Evolution. Music and Art (EvoMUSART2003), 2003, pp. 522–534.
[26] R. Mikic, D. Pecaric, and J. Pecaric, Inequalities of the Jensen and Edmundson-Lah-Ribaric type for 3-convex functions with applications, J. Math. Inequal., to appear.
[27] D. Mouillot and A. Lepretre, Introduction of relative abundance distribution (RAD) indices, estimated from the rank-frequency diagrams (RFD), to assess changes in community diversity, Envir. Monitor. Assessment., 63(2) (2000), 279–295.
[28] J. Pecaric, I. Peric, and G. Roquia, Exponentially convex functions generated by Wulbert’s inequality and Stolarsky-type means, Math. Comp. Model., 55, (2012), 1849–1857.
[29] J. E. Pecaric, F. Proschan, and Y. L. Tong, Convex functions, partial orderings and statistical applications, Academic Press Inc., San Diego 1992.
[30] M. Sababheh, Improved Jensen’s inequality, Math. Inequal. Appl., 20(2) (2017), 389–403.
[31] Z. K. Silagadze, Citations and the Zipf–Mandelbrot law Complex Syst., 11 (1997), 487–499.
[32] G. K. Zipf, The psychobiology of language, Cambridge, Houghton-Mifflin, 1935.
[33] G. K. Zipf, Human behavior and the principle of least effort, Reading, Addison-Wesley, 1949.