Advances in Operator Theory

Strengthened converses of the Jensen and Edmundson-Lah-Ribarič inequalities

Mario Krnić, Rozarija Mikić, and Josip Pečarić

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In this paper, we give converses of the Jensen and Edmundson-Lah-Ribarič inequalities which are more accurate than the existing ones. These converses are given in a difference form and they rely on the recent refinement of the Jensen inequality obtained via linear interpolation of a convex function. As an application, we also derive improved converse relations for generalized means, for the Hölder and Hermite-Hadamard inequalities as well as for the inequalities of Giaccardi and Petrović.

Article information

Adv. Oper. Theory, Volume 1, Number 1 (2016), 104-122.

Received: 28 October 2016
Accepted: 18 November 2016
First available in Project Euclid: 4 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26D15: Inequalities for sums, series and integrals
Secondary: 47B38: Operators on function spaces (general) 26A51: Convexity, generalizations

positive linear functional convex function converse Jensen inequality Edmundson–Lah–Ribarič inequality Hölder inequality Hermite–Hadamard inequality


Krnić, Mario; Mikić, Rozarija; Pečarić, Josip. Strengthened converses of the Jensen and Edmundson-Lah-Ribarič inequalities. Adv. Oper. Theory 1 (2016), no. 1, 104--122. doi:10.22034/aot.1610.1040.

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  • P. R. Beesack and J. E. Pečarić, On the Jessen's inequality for convex functions, J. Math. Anal. Appl. 110 (1985), 536–552.
  • P.S. Bullen, D. S. Mitrinović, and P. M. Vasić, Means and Their Inequalities., Dordrecht-Boston-Lancaster-Tokyo: Reidel Publ., 1988.
  • P. S. Bullen, Error estimates for some elementary quadrature rules, Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 602–633 (1978), 97–103.
  • D. Choi, M. Krni\' c, and J. Pečari\' c, Improved Jensen-type inequalities via linear interpolation and applications, J. Math. Inequal. 11 (2017), 301–322.
  • G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, second edition, Cambridge University Press, Cambridge, 1967.
  • R. Jakšić and J. Pečarić, New converses of Jensen's and Lah–Ribarič's inequality II, J. Math. Inequal. 7 (2013), 617–645.
  • B. Jessen, Bemaerkinger om konvekse Funktioner og Uligheder imellem Middelvaerdier I, Matematisk Tidsskrift. B tematisk tidsskrift. B (1931), 84–95.
  • M. Klaričić Bakula, J. Pečarić, and J. Perić, On the converse Jensen inequality, Appl. Math. Comp. 218 (2012), 6566–6575.
  • P. Lah and M. Ribarič, Converse of Jensen's inequality for convex functions, Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 412–460 (1973), 201–205.
  • D. S. Mitrinović and I. B. Lacković, Hermite and convexity, Aeq. Math. 28 (1985), 229–232.
  • C. P. Niculescu and L. E. Persson, Convex Functions and Their Applications – A Contemporary Approach, Springer, New York, 2006.
  • C. P. Niculescu and L. E. Persson, Old and new on the Hermite–Hadamard inequality, Real Anal. Exchange 29 (2003/2004), 663–685.
  • J. Pečarić and J. Peri\' c, New improvement of the converse Jensen inequality, preprint.
  • J. E. Pečari\' c, F. Proschan, and Y. L. Tong, Convex functions, partial orderings and statistical applications, Academic Press Inc., San Diego, 1992.
  • M. Petrović, Sur une fonctionnelle, Publ. Math. Univ. Belgrade 1 (1932), 149–156.
  • A. W. Roberts and D. E. Varberg, Convex functions, Academic Press, New York-London, 1973.
  • P. M. Vasić and J. E. Pečarić, On the Jensen inequality for monotone functions I, Anal. Univ. Timişoara Ser. Ştiinţ. Mat. 17 (1979), no. 1, 95–104.