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Connections Between the Jensen and the Chebychev Functionals

  • Flavia Corina MitroiEmail author
Part of the International Series of Numerical Mathematics book series (ISNM, volume 161)

Abstract

This work is devoted to the study of connections between the Jensen functional and the Chebychev functional for convex, superquadratic and strongly convex functions. We give a more general definition of these functionals and establish some inequalities involving them. The entire discussion incorporates both the discrete and the continuous approach.

Keywords

Jensen functional Chebychev functional Superquadratic Convex Strong convex function 

Mathematics Subject Classification

26B25 26E60 26D15 

Notes

Acknowledgements

The author is grateful to Professor S. Abramovich for her valuable suggestions. Thank the anonymous referee for his careful reading and pertinent comments. Also the author acknowledge the support of CNCSIS Grant 420/2008.

References

  1. 1.
    Abramovich, S., Dragomir, S.S.: Normalized Jensen functional, superquadracity and related inequalities. Int. Ser. Numer. Math. 157, 217–228 (2009) MathSciNetGoogle Scholar
  2. 2.
    Dragomir, S.S., Ionescu, N.M.: Some converse of Jensen’s inequality and applications. Anal. Numer. Theor. Approx. 23, 71–78 (1994) MathSciNetzbMATHGoogle Scholar
  3. 3.
    Dragomir, S.S.: Bounds for the normalised Jensen functional. Bull. Aust. Math. Soc. 74, 471–478 (2006) zbMATHCrossRefGoogle Scholar
  4. 4.
    Hiriart-Urruty, J.B., Lemaréchal, C.: Fundamentals of Convex Analysis. Springer, Berlin Heidelberg (2001) zbMATHCrossRefGoogle Scholar
  5. 5.
    Merentes, N., Nikodem, K.: Remarks on strongly convex functions. Aequ. Math. 80, 193–199 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Mitroi, F.-C.: Estimating the normalized Jensen functional. J. Math. Inequal. (to appear) Google Scholar
  7. 7.
    Niculescu, C.P.: An extension of Chebyshev’s inequality and its connection with Jensen’s inequality. J. Inequal. Appl. 6, 451–462 (2001) MathSciNetzbMATHGoogle Scholar
  8. 8.
    Polyak, B.T.: Existence theorems and convergence of minimizing sequences in extremum problems with restrictions. Sov. Math. Dokl. 7, 72–75 (1966) Google Scholar
  9. 9.
    Rajba, T., Wąsowicz, Sz.: Probabilistic characterization of strong convexity. Opusc. Math. (to appear) Google Scholar
  10. 10.
    Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14(5), 877–898 (1976) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CraiovaCraiovaRomania

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