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SUPERADDITIVITY OF SOME FUNCTIONALS ASSOCIATED WITH JENSEN’S INEQUALITY FOR CONVEX FUNCTIONS ON LINEAR SPACES WITH APPLICATIONS

Part of: Inequalities

Published online by Cambridge University Press:  02 March 2010

S. S. DRAGOMIR
S. S. DRAGOMIR*
Affiliation:
Research Group in Mathematical Inequalities and Applications, School of Engineering and Science, Victoria University, PO Box 14428, Melbourne, Victoria 8001, Australia (email: [email protected])
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Abstract

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Some new results related to Jensen’s celebrated inequality for convex functions defined on convex sets in linear spaces are given. Applications for norm inequalities in normed linear spaces and f-divergences in information theory are provided as well.

Keywords

convex functionsJensen’s inequalitynormsmean f-deviationf-divergences

MSC classification

Secondary: 26D15: Inequalities for sums, series and integrals
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 82 , Issue 1 , August 2010 , pp. 44 - 61
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

[1]Agarwal, R. P. and Dragomir, S. S., ‘The property of supermultiplicity for some classical inequalities and applications’, Comput. Math. Appl. 35(6) (1998), 105–118.CrossRefGoogle Scholar
[2]Beran, R., ‘Minimum Hellinger distance estimates for parametric models’, Ann. Statist. 5 (1977), 445–463.CrossRefGoogle Scholar
[3]Csiszár, I., ‘Information-type measures of differences of probability distributions and indirect observations’, Studia Sci. Math. Hungar. 2 (1967), 299–318.Google Scholar
[4]Csiszár, I. and Körner, J., Information Theory: Coding Theorems for Discrete Memoryless Systems (Academic Press, New York, 1981).Google Scholar
[5]Dragomir, S. S., ‘An improvement of Jensen’s inequality’, Bull. Math. Soc. Sci. Math. Roumanie 34(4) (1990), 291–296.Google Scholar
[6]Dragomir, S. S., ‘Some refinements of Ky Fan’s inequality’, J. Math. Anal. Appl. 163(2) (1992), 317–321.CrossRefGoogle Scholar
[7]Dragomir, S. S., ‘Some refinements of Jensen’s inequality’, J. Math. Anal. Appl. 168(2) (1992), 518–522.CrossRefGoogle Scholar
[8]Dragomir, S. S., ‘A further improvement of Jensen’s inequality’, Tamkang J. Math. 25(1) (1994), 29–36.CrossRefGoogle Scholar
[9]Dragomir, S. S., ‘A new improvement of Jensen’s inequality’, Indian J. Pure Appl. Math. 26(10) (1995), 959–968.Google Scholar
[10]Dragomir, S. S., ‘Bounds for the normalized Jensen functional’, Bull. Aust. Math. Soc. 74(3) (2006), 471–478.CrossRefGoogle Scholar
[11]Dragomir, S. S., ‘A refinement of Jensen’s inequality with applications for f-divergence measures’, Res. Rep. Coll. 10 (2007), Preprint; Supp., Article 15.Google Scholar
[12]Dragomir, S. S. and Goh, C. J., ‘A counterpart of Jensen’s discrete inequality for differentiable convex mappings and applications in information theory’, Math. Comput. Modelling 24(2) (1996), 1–11.CrossRefGoogle Scholar
[13]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
[14]Dragomir, S. S., Pečarić, J. and Persson, L. E., ‘Properties of some functionals related to Jensen’s inequality’, Acta Math. Hungar. 70(1–2) (1996), 129–143.CrossRefGoogle Scholar
[15]Kapur, J. N., ‘A comparative assessment of various measures of directed divergence’, Adv. Manag. Stud. 3(1) (1984), 1–16.Google Scholar
[16]Kato, M., Saito, K.-S. and Tamura, T., ‘Sharp triangle inequality and its reverse in Banach spaces’, Math. Inequal. Appl. 10(2) (2007), 451–460.Google Scholar
[17]Kullback, S., Information Theory and Statistics (Wiley, New York, 1959).Google Scholar
[18]Kullback, S. and Leibler, R. A., ‘On information and sufficiency’, Ann. Math. Statist. 22 (1951), 79–86.CrossRefGoogle Scholar
[19]Liese, F. and Vajda, I., Convex Statistical Distances (Teubner, Leipzig, 1987).Google Scholar
[20]Pečarić, J. and Dragomir, S. S., ‘A refinement of Jensen inequality and applications’, Studia Univ. Babeş-Bolyai Math. 24(1) (1989), 15–19.Google Scholar
[21]Rényi, A., ‘On measures of entropy and information’, in: Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, (ed. Neyman, J.) (University of California Press, Berkeley, CA, 1961).Google Scholar
[22]Topsoe, F., ‘Some inequalities for information divergence and related measures of discrimination’, Res. Rep. Coll. RGMIA 2(1) (1999), 85–98.7Google Scholar
[23]Vajda, I., Theory of Statistical Inference and Information (Kluwer, Boston, MA, 1989).Google Scholar
[24]Vasić, P. M. and Mijajlović, Ž., ‘On an index set function connected with Jensen inequality’, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 544–576 (1976), 110–112.Google Scholar