Best possible global bounds for Jensen functional

Simic, Slavko

doi:10.1090/S0002-9939-10-10353-0

Best possible global bounds for Jensen functional
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by Slavko Simic

Proc. Amer. Math. Soc. 138 (2010), 2457-2462

DOI: https://doi.org/10.1090/S0002-9939-10-10353-0

Published electronically: March 10, 2010

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Abstract:

We determine the form of the best possible global bounds for the Jensen functional on the real line. Thereby, previous results on this topic are essentially improved. Some applications in Analysis and Information Theory are also given.

References

Paul R. Beesack and Josip E. Pečarić, On Jessen’s inequality for convex functions, J. Math. Anal. Appl. 110 (1985), no. 2, 536–552. MR 805275, DOI 10.1016/0022-247X(85)90315-4
I. Budimir, S. S. Dragomir, and J. Pečarić, Further reverse results for Jensen’s discrete inequality and applications in information theory, JIPAM. J. Inequal. Pure Appl. Math. 2 (2001), no. 1, Article 5, 14. MR 1822298
Sever S. Dragomir, Bounds for the normalised Jensen functional, Bull. Austral. Math. Soc. 74 (2006), no. 3, 471–478. MR 2273755, DOI 10.1017/S000497270004051X
S. S. Dragomir, A converse result for Jensen’s discrete inequality via Gruss’ inequality and applications in information theory, An. Univ. Oradea Fasc. Mat. 7 (1999/00), 178–189. MR 1774897
S. S. Dragomir and C. J. Goh, Some bounds on entropy measures in information theory, Appl. Math. Lett. 10 (1997), no. 3, 23–28. MR 1457634, DOI 10.1016/S0893-9659(97)00028-1
Hardy G. H., Littlewood J. E., Pólya G.: Inequalities, Cambridge Univ. Press, Cambridge, 1988. MR0944909 (89d:26016)
D. S. Mitrinović, Analytic inequalities, Die Grundlehren der mathematischen Wissenschaften, Band 165, Springer-Verlag, New York-Berlin, 1970. In cooperation with P. M. Vasić. MR 0274686, DOI 10.1007/978-3-642-99970-3
Slavko Simic, Jensen’s inequality and new entropy bounds, Appl. Math. Lett. 22 (2009), no. 8, 1262–1265. MR 2532551, DOI 10.1016/j.aml.2009.01.040
Slavko Simic, On a global upper bound for Jensen’s inequality, J. Math. Anal. Appl. 343 (2008), no. 1, 414–419. MR 2412138, DOI 10.1016/j.jmaa.2008.01.060
Wilhelm Specht, Zur Theorie der elementaren Mittel, Math. Z. 74 (1960), 91–98 (German). MR 0117309, DOI 10.1007/BF01180475
Masaru Tominaga, Specht’s ratio in the Young inequality, Sci. Math. Jpn. 55 (2002), no. 3, 583–588. MR 1901045