The Jensen-Mercer Inequality with Infinite Convex Combinations

Year 2019, Volume: 7 Issue: 1, 19 - 27, 30.04.2019

Zlatko Pavic

https://doi.org/10.36753/mathenot.559241

Cited By: 4

Abstract

The paper deals with discrete forms of double inequalities related to convex functions of one variable.

Infinite convex combinations and sequences of convex combinations are included. The double inequality

form of the Jensen-Mercer inequality and its variants are especially studied.

Keywords

double inequality, infinite convex combination, Jensen-Mercer inequality

References

• [1] Hadamard, J., Étude sur les propriétés des fonctions entières et en particulier d’une fonction considerée par Riemann, J. Math. Pures Appl., 58(1893), 171-215.
• [2] Hermite, Ch., Sur deux limites d’une intégrale définie, Mathesis, 3(1883), 82.
• [3] Iveli´c, S., Matkovic, A. and Pecaric, J. E., On a Jensen-Mercer operator inequality, Banach J. Math. Anal., 5(2011), 19-28.
• [4] Jensen, J. L.W. V., Om konvekse Funktioner og Uligheder mellem Middelværdier, Nyt Tidsskr. Math. B, 16(1905), 49-68.
• [5] Khan, M. A., Khan, G. A., Jameel, M., Khan, K. A. and Kilicman, A., New refinements of Jensen-Mercer’s inequality J. Comput. Theor. Nanosci., 12(2015), 4442-4449.
• [6] Matkovic, A., Peˇcari´c, J. and Peri´c, I., A variant of Jensen’s inequality of Mercer’s type for operators with applications, Linear Algebra Appl., 418(2006), 551-564.
• [7] Mercer, A. McD., A variant of Jensen’s inequality, JIPAM, 4(2003), Article 73.
• [8] Niezgoda, M., A generalization of Mercer’s result on convex functions, Nonlinear Anal., 71(2009), 2771-2779.
• [9] Pavic, Z., Convex function and its secant, Adv. Inequal. Appl., 2015(2015), Article 5.
• [10] Pavic, Z., Generalizations of Jensen-Mercer’s inequality, J. Pure Appl. Math. Adv. Appl., 11(2014), 19-36.
• [11] Pavic Z., Geometric and analytic connections of the Jensen and Hermite-Hadamard inequality, Math. Sci. Appl. E-Notes, 4(2016), 69-76.
• [12] Pavic, Z., Inequalities with infinite convex combinations, submitted to FILOMAT.