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

  • Silvestru Sever DragomirEmail author
Part of the Springer Optimization and Its Applications book series (SOIA, volume 134)


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.


  1. 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. 2.
    M. Beth Bassat, f-entropies, probability of error and feature selection. Inf. Control 39, 227–242 (1978)Google Scholar
  3. 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)MathSciNetzbMATHGoogle Scholar
  4. 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)MathSciNetCrossRefGoogle Scholar
  5. 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).
  6. 6.
    C.H. Chen, Statistical Pattern Recognition (Hoyderc Book Co., Rocelle Park, 1973)Google Scholar
  7. 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. 8.
    C.K. Chow, C.N. Lin, Approximating discrete probability distributions with dependence trees. IEEE Trans. Inf. Theory 14(3), 462–467 (1968)CrossRefGoogle Scholar
  9. 9.
    I. Csiszár, Information-type measures of difference of probability distributions and indirect observations. Studia Math. Hung. 2, 299–318 (1967)MathSciNetzbMATHGoogle Scholar
  10. 10.
    I. Csiszár, On topological properties of f-divergences. Studia Math. Hung. 2, 329–339 (1967)zbMATHGoogle Scholar
  11. 11.
    I. Csiszár, J. Körner, Information Theory: Coding Theorem for Discrete Memoryless Systems (Academic, New York, 1981)zbMATHGoogle Scholar
  12. 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)zbMATHGoogle Scholar
  13. 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. 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).
  15. 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).
  16. 16.
    S.S. Dragomir, Bounds for the normalized Jensen functional. Bull. Aust. Math. Soc. 74(3), 471–476 (2006)CrossRefGoogle Scholar
  17. 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).
  18. 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: 978-1-4614-1520-6Google Scholar
  19. 19.
    S.S. Dragomir, Operator Inequalities of Ostrowski and Trapezoidal Type. Springer Briefs in Mathematics (Springer, New York, 2012), x+112 pp. ISBN: 978-1-4614-1778-1Google Scholar
  20. 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).
  21. 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).
  22. 22.
    S.S. Dragomir, Jensen type weighted inequalities for functions of selfadjoint and unitary operators. Ital. J. Pure Appl. Math. 32, 247–264 (2014)MathSciNetzbMATHGoogle Scholar
  23. 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).
  24. 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).
  25. 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)MathSciNetzbMATHGoogle Scholar
  26. 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. 27.
    D.V. Gokhale, S. Kullback, Information in Contingency Tables (Marcel Decker, New York, 1978)zbMATHGoogle Scholar
  28. 28.
    J.H. Havrda, F. Charvat, Quantification method classification process: concept of structural α-entropy. Kybernetika 3, 30–35 (1967)MathSciNetzbMATHGoogle Scholar
  29. 29.
    E. Hellinger, Neue Bergrüirdung du Theorie quadratisher Formerus von uneudlichvieleu Veränderlicher. J. für Reine Augeur. Math. 36, 210–271 (1909)Google Scholar
  30. 30.
    G. Helmberg, Introduction to Spectral Theory in Hilbert Space (Wiley, New York, 1969)zbMATHGoogle Scholar
  31. 31.
    H. Jeffreys, An invariant form for the prior probability in estimating problems. Proc. R. Soc. Lond. A 186, 453–461 (1946)CrossRefGoogle Scholar
  32. 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)zbMATHGoogle Scholar
  33. 33.
    T. Kailath, The divergence and Bhattacharyya distance measures in signal selection. IEEE Trans. Commun. Technol. COM-15, 52–60 (1967)CrossRefGoogle Scholar
  34. 34.
    J.N. Kapur, A comparative assessment of various measures of directed divergence. Adv. Manag. Stud. 3, 1–16 (1984)Google Scholar
  35. 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)CrossRefGoogle Scholar
  36. 36.
    S. Kullback, R.A. Leibler, On information and sufficiency. Ann. Math. Stat. 22, 79–86 (1951)MathSciNetCrossRefGoogle Scholar
  37. 37.
    J. Lin, Divergence measures based on the Shannon entropy. IEEE Trans. Inf. Theory 37(1), 145–151 (1991)MathSciNetCrossRefGoogle Scholar
  38. 38.
    C.A. McCarthy, c p. Isr. J. Math. 5, 249–271 (1967)Google Scholar
  39. 39.
    M. Mei, The theory of genetic distance and evaluation of human races. Jpn. J. Hum. Genet. 23, 341–369 (1978)CrossRefGoogle Scholar
  40. 40.
    B. Mond, J. Pečarić, Convex inequalities in Hilbert space. Houst. J. Math. 19, 405–420 (1993)MathSciNetzbMATHGoogle Scholar
  41. 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)MathSciNetzbMATHGoogle Scholar
  42. 42.
    E.C. Pielou, Ecological Diversity (Wiley, New York, 1975)Google Scholar
  43. 43.
    C.R. Rao, Diversity and dissimilarity coefficients: a unified approach. Theor. Popul. Biol. 21, 24–43 (1982)MathSciNetCrossRefGoogle Scholar
  44. 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. 547–561Google Scholar
  45. 45.
    A.W. Roberts, D.E. Varberg, Convex Functions (Academic, New York, 1973)zbMATHGoogle Scholar
  46. 46.
    A. Sen, On Economic Inequality (Oxford University Press, London, 1973)CrossRefGoogle Scholar
  47. 47.
    B.D. Sharma, D.P. Mittal, New non-additive measures of relative information. J. Comb. Inf. Syst. Sci. 2(4), 122–132 (1977)MathSciNetzbMATHGoogle Scholar
  48. 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)CrossRefGoogle Scholar
  49. 49.
    S. Simić, On a global upper bound for Jensen’s inequality. J. Math. Anal. Appl. 343, 414–419 (2008)MathSciNetCrossRefGoogle Scholar
  50. 50.
    I.J. Taneja, Generalised information measures and their applications.
  51. 51.
    H. Theil, Economics and Information Theory (North-Holland, Amsterdam, 1967)Google Scholar
  52. 52.
    H. Theil, Statistical Decomposition Analysis (North-Holland, Amsterdam, 1972)zbMATHGoogle Scholar
  53. 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. 54.
    I. Vajda, Theory of Statistical Inference and Information (Kluwer Academic Publishers, Dordrecht, 1989)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematics, College of Engineering & ScienceVictoria UniversityMelbourne CityAustralia
  2. 2.DST-NRF Centre of Excellence in the Mathematical and Statistical Sciences, School of Computer Science and Applied MathematicsUniversity of the WitwatersrandJohannesburgSouth Africa

Personalised recommendations