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SOME REVERSES OF THE JENSEN INEQUALITY WITH APPLICATIONS

Published online by Cambridge University Press:  07 February 2013

S. S. DRAGOMIR
S. S. DRAGOMIR*
Affiliation:
Mathematics, School of Engineering & Science, Victoria University, PO Box 14428, Melbourne City, MC 8001, Australia email sever.dragomir@vu.edu.au School of Computational & Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa email sever.dragomir@vu.edu.au
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Abstract

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Two new reverses of the celebrated Jensen’s inequality for convex functions in the general setting of the Lebesgue integral, with applications to means, Hölder’s inequality and $f$-divergence measures in information theory, are given.

Keywords

Jensen’s inequalityHölder’s inequalitymeasurable functionsLebesgue integraldivergence measuresf-divergence measures
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 87 , Issue 2 , April 2013 , pp. 177 - 194
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc.

References

Ali, S. M. and Silvey, S. D., ‘A general class of coefficients of divergence of one distribution from another’, J. R. Stat. Soc. Ser. B 28 (1966), 131–142.Google Scholar
Ben-Bassat, M., ‘$f$-entropies, probability of error and feature selection’, Inform. Control 39 (1978), 227–242.CrossRefGoogle Scholar
Bhattacharyya, A., ‘On a measure of divergence between two statistical populations defined by their probability distributions’, Bull. Calcutta Math. Soc. 35 (1943), 99–109.Google Scholar
Burbea, I. and Rao, C. R., ‘On the convexity of some divergence measures based on entropy function’, IEEE Trans. Inform. Theory 28 (3) (1982), 489–495.CrossRefGoogle Scholar
Chen, C. H., Statistical Pattern Recognition (Hoyderc Book Co., Rocelle Park, New York, 1973).Google Scholar
Chow, C. K. and Lin, C. N., ‘Approximating discrete probability distributions with dependence trees’, IEEE Trans. Inform. Theory 14 (3) (1968), 462–467.CrossRefGoogle Scholar
Csiszár, I., ‘Information-type measures of difference of probability distributions and indirect observations’, Studia Math. Hungarica 2 (1967), 299–318.Google Scholar
Csiszár, I., ‘On topological properties of $f$-divergences’, Studia Math. Hungarica 2 (1967), 329–339.Google Scholar
Csiszár, I. and Körner, J., Information Theory: Coding Theorem for Discrete Memoryless Systems (Academic Press, New York, 1981).Google Scholar
Dragomir, S. S., ‘A converse result for Jensen’s discrete inequality via Gruss’ inequality and applications in information theory’, An. Univ. Oradea Fasc. Mat. 7 (1999/2000), 178–189.Google Scholar
Dragomir, S. S., ‘On a reverse of Jensen’s inequality for isotonic linear functionals’, J. Inequal. Pure Appl. Math. 2 (3) (2001), (Article 36).Google Scholar
Dragomir, S. S., ‘A Grüss type inequality for isotonic linear functionals and applications’, Demonstratio Math. 36 (3) (2003), 551–562.Google Scholar
Dragomir, S. S., ‘Bounds for the deviation of a function from the chord generated by its extremities’, Bull. Aust. Math. Soc. 78 (2) (2008), 225–248.CrossRefGoogle Scholar
Dragomir, S. S., ‘Bounds for the normalized Jensen functional’, Bull. Aust. Math. Soc. 74 (3) (2006), 471–476.CrossRefGoogle Scholar
Dragomir, S. S. and Ionescu, N. M., ‘Some converse of Jensen’s inequality and applications’, Rev. Anal. Numér. Théor. Approx. 23 (1) (1994), 71–78.Google Scholar
Gokhale, D. V. and Kullback, S., Information in Contingency Tables (Marcel Decker, New York, 1978).Google Scholar
Havrda, J. H. and Charvat, F., ‘Quantification method classification process: concept of structural $\alpha $-entropy’, Kybernetika 3 (1967), 30–35.Google Scholar
Hellinger, E., ‘Neue Begründung der Theorie quadratischer Formen von unendlichvielen Veränderlichen’, J. reine angew. Math. 36 (1909), 210–271.CrossRefGoogle Scholar
Jeffreys, H., ‘An invariant form for the prior probability in estimating problems’, Proc. R. Soc. Lond. A 186 (1946), 453–461.Google ScholarPubMed
Kadota, T. T. and Shepp, L. A., ‘On the best finite set of linear observables for discriminating two Gaussian signals’, IEEE Trans. Inform. Theory 13 (1967), 288–294.CrossRefGoogle Scholar
Kailath, T., ‘The divergence and Bhattacharyya distance measures in signal selection’, IEEE Trans. Comm. Technology. (1967), 52–60; (Vol COM-15).CrossRefGoogle Scholar
Kapur, J. N., ‘A comparative assessment of various measures of directed divergence’, Adv. Management Stud. 3 (1984), 1–16.Google Scholar
Kazakos, D. and Cotsidas, T., ‘A decision theory approach to the approximation of discrete probability densities’, IEEE Trans. Perform. Anal. Machine Intell. 1 (1980), 61–67.CrossRefGoogle Scholar
Kullback, S. and Leibler, R. A., ‘On information and sufficiency’, Annals Math. Statist. 22 (1951), 79–86.CrossRefGoogle Scholar
Lin, J., ‘Divergence measures based on the Shannon entropy’, IEEE Trans. Inform. Theory 37 (1) (1991), 145–151.CrossRefGoogle Scholar
Mei, M., ‘The theory of genetic distance and evaluation of human races’, Japan J. Human Genetics 23 (1978), 341–369.Google Scholar
Niculescu, C. P., ‘An extension of Chebyshev’s inequality and its connection with Jensen’s inequality’, J. Inequal. Appl. 6 (4) (2001), 451–462.Google Scholar
Pielou, E. C., Ecological Diversity (Wiley, New York, 1975).Google Scholar
Rao, C. R., ‘Diversity and dissimilarity coefficients: a unified approach’, Theoretic Population Biol. 21 (1982), 24–43.CrossRefGoogle Scholar
Rényi, A., ‘On measures of entropy and information’, Proc. Fourth Berkeley Symp. Math. Stat. Prob. 1 (1961), 547–561, (University of California Press).Google Scholar
Sen, A., On Economic Inequality (Oxford University Press, London, 1973).CrossRefGoogle Scholar
Sharma, B. D. and Mittal, D. P., ‘New nonadditive measures of relative information’, J. Comb. Inf. Syst. Sci. 2 (4) (1977), 122–132.Google Scholar
Shioya, H. and Da-Te, T., ‘A generalisation of Lin divergence and the derivative of a new information divergence’, Elec. and Comm. in Japan 78 (7) (1995), 37–40.CrossRefGoogle Scholar
Simic, S., ‘On a global upper bound for Jensen’s inequality’, J. Math. Anal. Appl. 343 (2008), 414–419.CrossRefGoogle Scholar
Taneja, I. J., Generalised Information Measures and their Applications, http://www.mtm.ufsc.br/~taneja/bhtml/bhtml.html.Google Scholar
Theil, H., Economics and Information Theory (North-Holland, Amsterdam, 1967).Google Scholar
Theil, H., Statistical Decomposition Analysis (North-Holland, Amsterdam, 1972).Google Scholar
Topsoe, F., ‘Some inequalities for information divergence and related measures of discrimination’, Res. Rep. Coll., RGMIA 2 (1) (1999), 85–98.Google Scholar
Vajda, I., Theory of Statistical Inference and Information (Kluwer Academic Publishers, Dordrecht-Boston, 1989).Google Scholar