Weighted Steffensen Type Inequalities Involving Convex Functions

Abstract

The object is to obtain weighted Steffensen type inequalities for the class of convex functions using inequalities for the class of functions that are "convex at point $c$''. Additionally, we give weaker conditions for obtained weighted Steffensen type inequalities. Moreover, by further generalizations of these inequalities we obtain refined and sharpened versions.

Keywords

• Steffensen's inequality,measure theory,generalizations,convex function

References

1. [1] S. N. Bernstein, Sur les fonctions absolument monotones, Acta Math. Vol:52 (1929), 1–66.
2. [2] J. Jaksetic, J. Pecaric, K. Smoljak Kalamir, Measure theoretic generalization of Peˇcari´c, Mercer and Wu-Srivastava results, Sarajevo J. Math. Vol:12, No.24 (2016), 33–49.
3. [3] Z. Liu, On extension of Steffensen’s inequality, J. Math. Anal. Approx. Theory Vol:2, No.2 (2007), 132–139.
4. [4] P. R. Mercer, Extensions of Steffensen’s inequality, J. Math. Anal. Appl. Vol:246, No.1 (2000), 325–329.
5. [5] J. Pecaric Notes on some general inequalities, Publ. Inst. Math. (Beograd), Nouvelle serie Vol:32, No.46 (1982), 131–135.
6. [6] J. Pecaric, A. Perusic, K. Smoljak, Mercer and Wu-Srivastava generalisations of Steffensen’s inequality, Appl. Math. Comput. Vol:219, No.21 (2013), 10548–10558.
7. [7] J. Pecaric, K. Smoljak Kalamir, Generalized Steffensen type inequalities involving convex functions, J. Funct. Spaces Vol:2014, Article ID 428030, 10 pages.
8. [8] J. Pecaric, K. Smoljak, Steffensen type inequalities involving convex functions, Math. Inequal. Appl. Vol:18, No.1 (2015), 363–378.

9. [9] J. Pecaric, K. Smoljak Kalamir, S. Varosanec, Steffensen’s and related inequalities (A comprehensive survey and recent advances), Monograhps in inequalities 7, Element, Zagreb, 2014.
10. [10] J. F. Steffensen, On certain inequalities between mean values and their application to actuarial problems, Skand. Aktuarietids. (1918), 82–97.
11. [11] S. H. Wu, H. M. Srivastava, Some improvements and generalizations of Steffensen’s integral inequality, Appl. Math. Comput. Vol:192 (2007), 422-428.