We obtain new refinements of Jessen’s functional defined by means of positive linear functionals. The accumulated results are applied to weighted generalized and power means. We also obtain new refinements of numerous classical inequalities, such as the arithmetic-geometric mean inequality, Young’s inequality, and Hölder’s inequality.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. M. Aldaz, “A refinement of the inequality between arithmetic and geometric means,” J. Math. Inequal., 4, No. 2, 473–477 (2008).
J. M. Aldaz, “A stability version of Hölder’s inequality,” J. Math. Anal. Appl., 2, No. 343, 842–852 (2008).
J. M. Aldaz, “Self-improvement of the inequality between arithmetic and geometric means,” J. Math. Inequal., 2, No. 3, 213–216 (2009).
J. M. Aldaz, “A measure-theoretic version of the Dragomir–Jensen inequality,” Proc. Amer. Math. Soc., 140, No. 7, 2391–2399 (2012).
J. M. Aldaz, “Concentration of the ratio between the geometric and arithmetic means,” J. Theor. Probab., 23, No. 2, 498–508 (2010).
J. M. Aldaz, “Comparison of differences between arithmetic and geometric means,” Tamkang J. Math., 42, No. 4, 453–462 (2011).
P. R. Beesack and J. E. Pečarić, “On Jessen’s inequality for convex functions,” J. Math. Anal. Appl., 2, No. 110, 536–552 (1985).
S. S. Dragomir, C. E. M. Pearce, and J. E. Pečarić, “On Jessen’s and related inequalities for isotonic sublinear functionals,” Acta. Sci. Math. (Szeged), 61, No. 1-4, 373–382 (1995).
S. S. Dragomir, J. E. Pečarić, L. E. Persson, “Properties of some functionals related to Jensen’s inequality,” Acta Math. Hung., 70, No. 1-2, 129–143 (1996).
F. V. Husseinov, “On Jensen’s inequality,” Mat. Zametki, 41, 798–806 (1987).
M. Krnić, N. Lovričević, and J. Pečarić, “Jessen’s functional, its properties, and applications,” An. Sti. Univ. Ovidius Constanta, Ser. Mat., 20, No. 1, 225–248 (2012).
M. Krnić, N. Lovričević, and J. Pečarić, “On some properties of Jensen–Steffensen’s functional,” An. Univ. Craiova. Mat. Inform., 38, No. 2, 43–54 (2011).
D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer AP, Dordrecht, etc. (1993).
J. E. Pečarić and P. R. Beesack, “On Jessen’s inequality for convex functions II,” J. Math. Anal. Appl., 118, No. 1, 125–144 (1986).
J. E. Pečarić, “On Jessen’s inequality for convex functions III,” J. Math. Anal. Appl., 156, No. 1, 231–239 (1991).
J. E. Pečarić and I. Rasa, “On Jessen’s inequality,” Acta. Sci. Math. (Szeged), 56, No. 3-4, 305–309 (1992).
J. E. Pečarić, F. Proschan, and Y. L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, Academic Press (1992).
I. Rasa, “A note on Jessen’s inequality,” Itin. Semin. Funct. Equat. Approxim. Conv. Cluj-Napoca, Univ. “Babes-Bolyai”, 275–280 (1988).
Author information
Authors and Affiliations
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 68, No. 7, pp. 879–896, July, 2016.
Rights and permissions
About this article
Cite this article
Barbir, A., Himmelreich, K.K. & Pečarić, J. Refinements of Jessen’s Functional. Ukr Math J 68, 1000–1020 (2016). https://doi.org/10.1007/s11253-016-1273-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-016-1273-7