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.
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References
Abramovich, S., Dragomir, S.S.: Normalized Jensen functional, superquadracity and related inequalities. Int. Ser. Numer. Math. 157, 217–228 (2009)
Dragomir, S.S., Ionescu, N.M.: Some converse of Jensen’s inequality and applications. Anal. Numer. Theor. Approx. 23, 71–78 (1994)
Dragomir, S.S.: Bounds for the normalised Jensen functional. Bull. Aust. Math. Soc. 74, 471–478 (2006)
Hiriart-Urruty, J.B., Lemaréchal, C.: Fundamentals of Convex Analysis. Springer, Berlin Heidelberg (2001)
Merentes, N., Nikodem, K.: Remarks on strongly convex functions. Aequ. Math. 80, 193–199 (2010)
Mitroi, F.-C.: Estimating the normalized Jensen functional. J. Math. Inequal. (to appear)
Niculescu, C.P.: An extension of Chebyshev’s inequality and its connection with Jensen’s inequality. J. Inequal. Appl. 6, 451–462 (2001)
Polyak, B.T.: Existence theorems and convergence of minimizing sequences in extremum problems with restrictions. Sov. Math. Dokl. 7, 72–75 (1966)
Rajba, T., Wąsowicz, Sz.: Probabilistic characterization of strong convexity. Opusc. Math. (to appear)
Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14(5), 877–898 (1976)
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.
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Mitroi, F.C. (2012). Connections Between the Jensen and the Chebychev Functionals. In: Bandle, C., Gilányi, A., Losonczi, L., Plum, M. (eds) Inequalities and Applications 2010. International Series of Numerical Mathematics, vol 161. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0249-9_17
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DOI: https://doi.org/10.1007/978-3-0348-0249-9_17
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