A Jensen–Sherman type inequality with a control map for G-invariant convex functions


In this paper, the standard Jensen inequality for two points and convex functions is generalized to a version for four points satisfying Sherman’s majorization condition and for G-invariant convex functions, where G is a subgroup of the orthogonal group acting on an inner product space. A control map is used. A comparison of the obtained inequality and Jensen inequality is presented.

This is a preview of subscription content, log in to check access.


  1. 1.

    Adil Khan, M., Ivelić Bradanović, S., Pečarić, J.: On Sherman’s type inequalities for \(n\)-convex function with applications. Konuralp J. Math. 4, 255–260 (2016)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Adil Khan, M., Khan, J., Pečarić, J.: Generalizations of Sherman’s inequality by Montgomery identity and Green function. Electron. J. Math. Analysis Appl. 5(1), 1–16 (2017)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Agarwal, R.P., Ivelić Bradanović, S., Pečarić, J.: Generalizations of Sherman’s inequality by Lidstone’s interpolating polynomial. J. Inequal. Appl. 2016(6), 18 (2016). https://doi.org/10.1186/s13660-015-0935-6

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Burtea, A.-M.: Two examples of weighted majorization. Ann. Univ. Craiova Ser. Math. Inf. 37, 92–99 (2010)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Eaton, M. L.: On group induced orderings, monotone functions, and convolution theorems. In: Tong Y. L. (ed.) Inequalities in Statistics and Probability, IMS Lectures Notes Monogr. Ser., vol. 5, pp. 13–25 (1984)

  6. 6.

    Eaton, M.L.: Group induced orderings with some applications in statistics. CWI Newsl. 16, 3–31 (1987)

    MathSciNet  Google Scholar 

  7. 7.

    Hardy, G.M., Littlewood, J.E., Pólya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1952)

    Google Scholar 

  8. 8.

    Ivelić Bradanović, S., Latif, N., Pečarić, J.: Generalizations of Sherman’s theorem by Taylor’s formula. J. Inequal. Spec. Funct. 8, 18–30 (2017)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Ivelić Bradanović, S., Pečarić, J.: Generalizations of Sherman’s inequality. Period. Math. Hung. 74, 197–219 (2017)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Karamata, J.: Sur une inégalité rélative aux fonctions convexes. Publ. Math. Univ. Belgrade 1, 145–148 (1932)

    MATH  Google Scholar 

  11. 11.

    Marshall, A.W., Olkin, I., Arnold, B.C.: Inequalities: Theory of Majorization and Its Applications, 2nd edn. Springer, New York (2011)

    Google Scholar 

  12. 12.

    Niezgoda, M.: Remarks on Sherman like inequalities for \((\alpha,\beta )\)-convex functions. Math. Inequal. Appl. 17, 1579–1590 (2014)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Niezgoda, M.: On Sherman method to deriving inequalities for some classes of functions related to convexity. In: Agarwal, P., Dragomir, S.S., Jleli, M., Samet, B. (eds.) Advances in Mathematical Inequalities and Applications. Trends in mathematics, pp. 219–245. Springer, Birkhauser, Basel (2018)

    Google Scholar 

  14. 14.

    Niezgoda, M.: Nonlinear Sherman-type inequalities. Adv. Nonlinear Anal. 9(1), 168–175 (2020)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Niezgoda, M.: A majorization gradient inequality for symmetric convex functions. J. Convex Anal. 27(4), (2020)

  16. 16.

    Rado, R.: An inequality. J. London Math. Soc. 27, 1–6 (1952)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Sherman, S.: On a theorem of Hardy, Littlewood, Pólya, and Blackwell. Proc. Natl. Acad. Sci. USA 37, 826–831 (1951)

    MATH  Google Scholar 

  18. 18.

    Steerneman, A.G.M.: G-majorization, group-induced cone orderings and reflection groups. Linear Algebra Appl. 127, 107–119 (1990)

    MathSciNet  MATH  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Marek Niezgoda.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Niezgoda, M. A Jensen–Sherman type inequality with a control map for G-invariant convex functions. Positivity (2020). https://doi.org/10.1007/s11117-020-00769-3

Download citation


  • Convex function
  • Jensen inequality
  • Sherman inequality
  • G-majorization
  • Control map

2010 Mathematics Subject Classification

  • Primary 52A41
  • 26B25
  • Secondary 26D10
  • 15B51