Inequalities for Synchronous Functions and Applications

Abstract

Some inequalities for synchronous functions that are a mixture between Cebyšev’s and Jensen's inequality are provided. Applications for $f$ -divergence measure and some particular instances including Kullback-Leibler divergence, Jeffreys divergence and $\chi ^{2}$-divergence are also given.

Keywords

• Synchronous Functions,Lipschitzian functions,Jensen's inequality,$f$-divergence measure,Kullback-Leibler divergence,Jeffreys divergence measure

References

1. [1] P. Cerone and S. S. Dragomir, Approximation of the integral mean divergence and f-divergence via mean results. Math. Comput. Modelling 42 (2005), no. 1-2, 207–219.
2. [2] P. Cerone, S. S. Dragomir and F. Österreicher, Bounds on extended f-divergences for a variety of classes, Kybernetika (Prague) 40 (2004), no. 6, 745–756. Preprint, RGMIA Res. Rep. Coll. 6(2003), No.1, Article 5. [ONLINE: http://rgmia.vu.edu.au/v6n1.html].
3. [3] I. Csiszár, Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizität von Markoffschen Ketten. (German) Magyar Tud. Akad. Mat. Kutató Int. Közl. 8 (1963) 85–108.
4. [4] S. S. Dragomir, Some inequalities for (m;M)-convex mappings and applications for the Csiszár $\Phi$-divergence in information theory. Math. J. Ibaraki Univ. 33 (2001), 35–50.
5. [5] S. S. Dragomir, Some inequalities for two Csiszár divergences and applications. Mat. Bilten No. 25 (2001), 73–90.
6. [6] S. S. Dragomir, An upper bound for the Csiszár f-divergence in terms of the variational distance and applications. Panamer. Math. J. 12 (2002), no. 4, 43–54.
7. [7] S. S. Dragomir, Upper and lower bounds for Csiszár f-divergence in terms of Hellinger discrimination and applications. Nonlinear Anal. Forum 7 (2002), no. 1, 1–13
8. [8] S. S. Dragomir, Bounds for f-divergences under likelihood ratio constraints. Appl. Math. 48 (2003), no. 3, 205–223.

9. [9] S. S. Dragomir, New inequalities for Csiszár divergence and applications. Acta Math. Vietnam. 28 (2003), no. 2, 123–134.
10. [10] S. S. Dragomir, A generalized f-divergence for probability vectors and applications. Panamer. Math. J. 13 (2003), no. 4, 61–69.
11. [11] S. S. Dragomir, Some inequalities for the Csiszár '-divergence when ' is an L-Lipschitzian function and applications. Ital. J. Pure Appl. Math. No. 15 (2004), 57–76.
12. [12] S. S. Dragomir, A converse inequality for the Csiszár $\Phi$-divergence. Tamsui Oxf. J. Math. Sci. 20 (2004), no. 1, 35–53.
13. [13] S. S. Dragomir, Some general divergence measures for probability distributions. Acta Math. Hungar. 109 (2005), no. 4, 331–345.
14. [14] S. S. Dragomir, A refinement of Jensen’s inequality with applications for f-divergence measures. Taiwanese J. Math. 14 (2010), no. 1, 153–164.
15. [15] S. S. Dragomir, A generalization of f-divergence measure to convex functions defined on linear spaces. Commun. Math. Anal. 15 (2013), no. 2, 1–14.
16. [16] H. Jeffreys, Theory of Probability, Oxford University Press, 1948, 2nd ed.
17. [17] F. Liese and I. Vajda, Convex Statistical Distances, Teubuer-Texte zur Mathematik, Band 95, Leipzig, 1987.