Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-15T19:19:20.538Z Has data issue: false hasContentIssue false
  • English
  • Français

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])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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