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Reverses of Jensen’s Integral Inequality and Applications: A Survey of Recent Results

Part of the Springer Optimization and Its Applications book series (SOIA,volume 134)

Abstract

Several new reverses of the celebrated Jensen’s inequality for convex functions and Lebesgue integral on measurable spaces are surveyed. Applications for weighted discrete means, to Hölder inequality, Cauchy-Bunyakovsky-Schwarz inequality and for f-divergence measures in information theory are also given. Finally, applications for functions of selfadjoint operators in Hilbert spaces with some examples of interest are also provided.

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References

  1. S.M. Ali, S.D. Silvey, A general class of coefficients of divergence of one distribution from another. J. R. Stat. Soc. Ser. B 28, 131–142 (1966)

    Google Scholar 

  2. M. Beth Bassat, f-entropies, probability of error and feature selection. Inf. Control 39, 227–242 (1978)

    Google Scholar 

  3. A. Bhattacharyya, On a measure of divergence between two statistical populations defined by their probability distributions. Bull. Calcutta Math. Soc. 35, 99–109 (1943)

    MathSciNet  MATH  Google Scholar 

  4. I. Burbea, C.R. Rao, On the convexity of some divergence measures based on entropy function. IEEE Trans. Inf. Theory 28(3), 489–495 (1982)

    MathSciNet  CrossRef  Google Scholar 

  5. P. Cerone, S.S. Dragomir, A refinement of the Grüss inequality and applications. Tamkang J. Math. 38(1), 37–49 (2007). Preprint RGMIA Res. Rep. Coll. 5(2), Article 14 (2002). http://rgmia.org/papers/v5n2/RGIApp.pdf

  6. C.H. Chen, Statistical Pattern Recognition (Hoyderc Book Co., Rocelle Park, 1973)

    Google Scholar 

  7. X.L. Cheng, J. Sun, A note on the perturbed trapezoid inequality. J. Inequal. Pure Appl. Math. 3(2), Article 29 (2002)

    Google Scholar 

  8. C.K. Chow, C.N. Lin, Approximating discrete probability distributions with dependence trees. IEEE Trans. Inf. Theory 14(3), 462–467 (1968)

    CrossRef  Google Scholar 

  9. I. Csiszár, Information-type measures of difference of probability distributions and indirect observations. Studia Math. Hung. 2, 299–318 (1967)

    MathSciNet  MATH  Google Scholar 

  10. I. Csiszár, On topological properties of f-divergences. Studia Math. Hung. 2, 329–339 (1967)

    MATH  Google Scholar 

  11. I. Csiszár, J. Körner, Information Theory: Coding Theorem for Discrete Memoryless Systems (Academic, New York, 1981)

    MATH  Google Scholar 

  12. S.S. Dragomir, A converse result for Jensen’s discrete inequality via Grüss’ inequality and applications in information theory. An. Univ. Oradea Fasc. Mat. 7, 178–189 (1999/2000)

    MATH  Google Scholar 

  13. S.S. Dragomir, On a reverse of Jessen’s inequality for isotonic linear functionals. J. Inequal. Pure Appl. Math. 2(3), Article 36 (2001)

    Google Scholar 

  14. S.S. Dragomir, A Grüss type inequality for isotonic linear functionals and applications. Demonstratio Math. 36(3), 551–562 (2003). Preprint RGMIA Res. Rep. Coll. 5(Supplement), Article 12 (2002). http://rgmia.org/papers/v5e/GTIILFApp.pdf

  15. S.S. Dragomir, A converse inequality for the Csiszár Φ-divergence. Tamsui Oxf. J. Math. Sci. 20(1), 35–53 (2004). Preprint in S.S. Dragomir (ed.), Inequalities for Csiszár f-Divergence in Information Theory. RGMIA Monographs (Victoria University, 2000). http://rgmia.org/papers/Csiszar/Csiszar.pdf

  16. S.S. Dragomir, Bounds for the normalized Jensen functional. Bull. Aust. Math. Soc. 74(3), 471–476 (2006)

    CrossRef  Google Scholar 

  17. S.S. Dragomir, Superadditivity of the Jensen integral inequality with applications. Miskolc Math. Notes 13(2), 303–316 (2012). Preprint RGMIA Res. Rep. Coll. 14, Article 75 (2011). http://rgmia.org/papers/v14/v14a75.pdf

  18. S.S. Dragomir, Operator Inequalities of the Jensen, Čebyšev and Grüss Type. Springer Briefs in Mathematics (Springer, New York, 2012), xii+121 pp. ISBN: 978-1-4614-1520-6

    Google Scholar 

  19. S.S. Dragomir, Operator Inequalities of Ostrowski and Trapezoidal Type. Springer Briefs in Mathematics (Springer, New York, 2012), x+112 pp. ISBN: 978-1-4614-1778-1

    Google Scholar 

  20. S.S. Dragomir, Some reverses of the Jensen inequality with applications. Bull. Aust. Math. Soc. 87(2), 177–194 (2013). Preprint RGMIA Res. Rep. Coll. 14, Article 72 (2011). http://rgmia.org/papers/v14/v14a72.pdf

  21. S.S. Dragomir, Reverses of the Jensen inequality in terms of first derivative and applications. Acta Math. Vietnam. 38(3), 429–446 (2013). Preprint RGMIA Res. Rep. Coll. 14, Article 71 (2011). http://rgmia.org/papers/v14/v14a71.pdf

  22. S.S. Dragomir, Jensen type weighted inequalities for functions of selfadjoint and unitary operators. Ital. J. Pure Appl. Math. 32, 247–264 (2014)

    MathSciNet  MATH  Google Scholar 

  23. S.S. Dragomir, A refinement and a divided difference reverse of Jensen’s inequality with applications. Rev. Colomb. Mat. 50(1), 17–39 (2016). Preprint RGMIA Res. Rep. Coll. 14, Article 74 (2011). http://rgmia.org/papers/v14/v14a74.pdf

  24. S.S. Dragomir, Weighted reverse inequalities of Jensen type for functions of selfadjoint operators. Transylv. J. Math. Mech. 8(1), 29–44 (2016). Preprint RGMIA Res. Rep. Coll. 18, Article 110 (2015). http://rgmia.org/papers/v18/v18a110.pdf

  25. S.S. Dragomir, N.M. Ionescu, Some converse of Jensen’s inequality and applications. Rev. Anal. Numér. Théor. Approx. 23(1), 71–78 (1994)

    MathSciNet  MATH  Google Scholar 

  26. T. Furuta, J. Mićić Hot, J. Pečarić, Y. Seo, Mond-Pečarić Method in Operator Inequalities. Inequalities for Bounded Selfadjoint Operators on a Hilbert Space (Element, Zagreb, 2005)

    Google Scholar 

  27. D.V. Gokhale, S. Kullback, Information in Contingency Tables (Marcel Decker, New York, 1978)

    MATH  Google Scholar 

  28. J.H. Havrda, F. Charvat, Quantification method classification process: concept of structural α-entropy. Kybernetika 3, 30–35 (1967)

    MathSciNet  MATH  Google Scholar 

  29. E. Hellinger, Neue Bergrüirdung du Theorie quadratisher Formerus von uneudlichvieleu Veränderlicher. J. für Reine Augeur. Math. 36, 210–271 (1909)

    Google Scholar 

  30. G. Helmberg, Introduction to Spectral Theory in Hilbert Space (Wiley, New York, 1969)

    MATH  Google Scholar 

  31. H. Jeffreys, An invariant form for the prior probability in estimating problems. Proc. R. Soc. Lond. A 186, 453–461 (1946)

    CrossRef  Google Scholar 

  32. T.T. Kadota, L.A. Shepp, On the best finite set of linear observables for discriminating two Gaussian signals. IEEE Trans. Inf. Theory 13, 288–294 (1967)

    MATH  Google Scholar 

  33. T. Kailath, The divergence and Bhattacharyya distance measures in signal selection. IEEE Trans. Commun. Technol. COM-15, 52–60 (1967)

    CrossRef  Google Scholar 

  34. J.N. Kapur, A comparative assessment of various measures of directed divergence. Adv. Manag. Stud. 3, 1–16 (1984)

    Google Scholar 

  35. D. Kazakos, T. Cotsidas, A decision theory approach to the approximation of discrete probability densities. IEEE Trans. Pattern Anal. Mach. Intell. 1, 61–67 (1980)

    CrossRef  Google Scholar 

  36. S. Kullback, R.A. Leibler, On information and sufficiency. Ann. Math. Stat. 22, 79–86 (1951)

    MathSciNet  CrossRef  Google Scholar 

  37. J. Lin, Divergence measures based on the Shannon entropy. IEEE Trans. Inf. Theory 37(1), 145–151 (1991)

    MathSciNet  CrossRef  Google Scholar 

  38. C.A. McCarthy, c p. Isr. J. Math. 5, 249–271 (1967)

    Google Scholar 

  39. M. Mei, The theory of genetic distance and evaluation of human races. Jpn. J. Hum. Genet. 23, 341–369 (1978)

    CrossRef  Google Scholar 

  40. B. Mond, J. Pečarić, Convex inequalities in Hilbert space. Houst. J. Math. 19, 405–420 (1993)

    MathSciNet  MATH  Google Scholar 

  41. C.P. Niculescu, An extension of Chebyshev’s inequality and its connection with Jensen’s inequality. J. Inequal. Appl. 6(4), 451–462 (2001)

    MathSciNet  MATH  Google Scholar 

  42. E.C. Pielou, Ecological Diversity (Wiley, New York, 1975)

    Google Scholar 

  43. C.R. Rao, Diversity and dissimilarity coefficients: a unified approach. Theor. Popul. Biol. 21, 24–43 (1982)

    MathSciNet  CrossRef  Google Scholar 

  44. A. Rényi, On measures of entropy and information, in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, vol. 1 (University of California Press, Berkeley, 1961), pp. 547–561

    Google Scholar 

  45. A.W. Roberts, D.E. Varberg, Convex Functions (Academic, New York, 1973)

    MATH  Google Scholar 

  46. A. Sen, On Economic Inequality (Oxford University Press, London, 1973)

    CrossRef  Google Scholar 

  47. B.D. Sharma, D.P. Mittal, New non-additive measures of relative information. J. Comb. Inf. Syst. Sci. 2(4), 122–132 (1977)

    MathSciNet  MATH  Google Scholar 

  48. H. Shioya, T. Da-Te, A generalisation of Lin divergence and the derivative of a new information divergence. Electron. Commun. Jpn. 78(7), 37–40 (1995)

    CrossRef  Google Scholar 

  49. S. Simić, On a global upper bound for Jensen’s inequality. J. Math. Anal. Appl. 343, 414–419 (2008)

    MathSciNet  CrossRef  Google Scholar 

  50. I.J. Taneja, Generalised information measures and their applications. http://www.mtm.ufsc.br/~taneja/bhtml/bhtml.html

  51. H. Theil, Economics and Information Theory (North-Holland, Amsterdam, 1967)

    Google Scholar 

  52. H. Theil, Statistical Decomposition Analysis (North-Holland, Amsterdam, 1972)

    MATH  Google Scholar 

  53. F. Topsoe, Some inequalities for information divergence and related measures of discrimination. Preprint RGMIA Res. Rep. Coll. 2(1), 85–98 (1999)

    Google Scholar 

  54. I. Vajda, Theory of Statistical Inference and Information (Kluwer Academic Publishers, Dordrecht, 1989)

    MATH  Google Scholar 

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Correspondence to Silvestru Sever Dragomir .

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Dragomir, S.S. (2018). Reverses of Jensen’s Integral Inequality and Applications: A Survey of Recent Results. In: Rassias, T. (eds) Applications of Nonlinear Analysis . Springer Optimization and Its Applications, vol 134. Springer, Cham. https://doi.org/10.1007/978-3-319-89815-5_8

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