Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-04T21:48:45.957Z Has data issue: false hasContentIssue false
  • English
  • Français

INEQUALITIES IN TERMS OF THE GÂTEAUX DERIVATIVES FOR CONVEX FUNCTIONS ON LINEAR SPACES WITH APPLICATIONS

Part of: Inequalities

Published online by Cambridge University Press:  07 February 2011

S. S. DRAGOMIR
S. S. DRAGOMIR*
Affiliation:
Mathematics, School of Engineering and Science, Victoria University, PO Box 14428 Melbourne City, MC 8001, Australia School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa (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 inequalities in terms of the Gâteaux derivatives related to Jensen’s inequality for convex functions defined on linear spaces are given. Applications for norms, mean f-deviations and f-divergence measures are provided as well.

Keywords

convex functionsGâteaux derivativesJensen’s inequalitynormsmeanf-deviationsf-divergence measures

MSC classification

Secondary: 26D15: Inequalities for sums, series and integrals
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 83 , Issue 3 , June 2011 , pp. 500 - 517
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Csiszár, I., ‘Information-type measures of differences of probability distributions and indirect observations’, Studia Sci. Math. Hungar. 2 (1967), 299–318.Google Scholar
[2]Csiszár, I. and Körner, J., Information Theory: Coding Theorems for Discrete Memoryless Systems (Academic Press, New York, 1981).Google Scholar
[3]Dragomir, S. S., ‘An improvement of Jensen’s inequality’, Bull. Math. Soc. Sci. Math. Roumanie 34(82) (1990), 291–296.Google Scholar
[4]Dragomir, S. S., ‘Some refinements of Ky Fan’s inequality’, J. Math. Anal. Appl. 163(2) (1992), 317–321.Google Scholar
[5]Dragomir, S. S., ‘Some refinements of Jensen’s inequality’, J. Math. Anal. Appl. 168(2) (1992), 518–522.Google Scholar
[6]Dragomir, S. S., ‘A further improvement of Jensen’s inequality’, Tamkang J. Math. 25(1) (1994), 29–36.Google Scholar
[7]Dragomir, S. S., ‘A new improvement of Jensen’s inequality’, Indian J. Pure Appl. Math. 26(10) (1995), 959–968.Google Scholar
[8]Dragomir, S. S., Semi-inner Products and Applications (Nova Science Publishers, New York, 2004).Google Scholar
[9]Dragomir, S. S., ‘A refinement of Jensen’s inequality with applications for f-divergence measures’, Taiwanese J. Math. 14(1) (2010), 153–164.Google Scholar
[10]Dragomir, S. S., ‘A new refinement of Jensen’s inequality in linear spaces with applications’, RGMIA Res. Rep. Coll. 12 (2009), Supplement, Article 6.http://www.staff.vu.edu.au/RGMIA/v12(E).asp.Google Scholar
[11]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.Google Scholar
[12]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
[13]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.Google Scholar
[14]Pečarić, J. and Dragomir, S. S., ‘A refinement of Jensen inequality and applications’, Stud. Univ. Babeş-Bolyai Math. 24(1) (1989), 15–19.Google Scholar