Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T17:56:19.179Z Has data issue: false hasContentIssue false
  • English
  • Français

BOUNDS FOR THE DEVIATION OF A FUNCTION FROM THE CHORD GENERATED BY ITS EXTREMITIES

Part of: Inequalities

Published online by Cambridge University Press:  01 October 2008

S. S. DRAGOMIR
S. S. DRAGOMIR*
Affiliation:
School of Computer Science and Mathematics, Victoria University, PO Box 14428, Melbourne City Mail Centre, Vic, 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.

Sharp bounds for the deviation of a real-valued function f defined on a compact interval [a,b] to the chord generated by its end points (a,f(a)) and (b,f(b)) under various assumptions for f and f′, including absolute continuity, convexity, bounded variation, and monotonicity, are given. Some applications for weighted means and f-divergence measures in information theory are also provided.

Keywords

bounded variationabsolute continuityconvexitymonotonicityweighted meansanalytic inequalitiesf-divergence measuresinformation theory

MSC classification

Secondary: 26D15: Inequalities for sums, series and integrals
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 78 , Issue 2 , October 2008 , pp. 225 - 248
Copyright
Copyright © 2008 Australian Mathematical Society

References

[1]Beran, R., ‘Minimum Hellinger distance estimates for parametric models’, Ann. Statist. 5 (1977), 445–463.CrossRefGoogle Scholar
[2]Beesack, P. R. and Pečarić, J. E., ‘On Jessen’s inequality for convex functions’, J. Math. Anal. Appl 110 (1985), 536–552.CrossRefGoogle Scholar
[3]Cerone, P., ‘On an identity for the Chebychev functional and some ramifications’, J. Ineq. Pure Appl. Math 3(1) (2002), Art. 4. [http://jipam.vu.edu.au/article.php?sid=157].Google Scholar
[4]Csiszár, I., ‘Information-type measures of differences of probability distributions and indirect observations’, Studia Sci. Math. Hungar. 2 (1967), 299–318.Google Scholar
[5]Csiszár, I. and Körner, J., Information Theory: Coding Theorems for Discrete Memoryless Systems (Academic Press, New York, 1981).Google Scholar
[6]Dragomir, S. S., Other inequalities for Csiszár divergence and applications, RGMIA Monographs, Victoria University, Preprint, 2000. [http://sci.vu.edu.au/∼rgmia/Csiszar/ICDApp.pdf].Google Scholar
[7]Dragomir, S. S., ‘A generalisation of Cerone’s identity and applications’, Oxford Tamsui J. Math. Sci. 23(1) (2007), 79–90. Preprint RGMIA Res. Rep. Coll., 8(2) (2005), Art. 19. [http://rgmia.vu.edu.au/v8n2.html].Google Scholar
[8]Dragomir, S. S., ‘Inequalities for Stieltjes integrals with convex integrators and applications’, Appl. Math. Lett. 20 (2007), 123–130.CrossRefGoogle Scholar
[9]Kapur, J. N., ‘A comparative assessment of various measures of directed divergence’, Adv. Management Studies 3(1) (1984), 1–16.Google Scholar
[10]Kullback, S. and Leibler, R. A., ‘On information and sufficiency’, Ann. Math. Statist. 22 (1951), 79–86.CrossRefGoogle Scholar
[11]Liese, F. and Vajda, I., Convex Statistical Distances (Teubner, Leipzig, 1987).Google Scholar
[12]Pečarić, J. E., Proschan, F. and Tong, Y. L., Convex Functions, Partial Orderings and Statistical Applications (Academic Press, Boston, MA, 1992).Google Scholar
[13]Rényi, A., ‘On measures of entropy and information’, in: Proc. Fourth Berkeley Symp. Math. Statist. Prob., 1 (University of California Press, Berkeley, CA, 1961).Google Scholar
[14]Topsøe, F., ‘Some inequalities for information divergence and related measures of discrimination’, Res. Rep. Coll. RGMIA 2(1) (1999), 85–98.Google Scholar
[15]Vajda, I., Theory of Statistical Inference and Information (Kluwer, Boston, MA, 1989).Google Scholar