In this paper, we consider the definition of “useful” Csiszár divergence and “useful” Zipf-Mandelbrot law associated with the real utility distribution to give the results for majorizatioQn inequalities by using monotonic sequences. We obtain the equivalent statements between continuous convex functions and Green functions via majorization inequalities, “useful” Csiszár functional and “useful” Zipf-Mandelbrot law. By considering “useful” Csiszár divergence in the integral case, we give the results for integral majorization inequality. Towards the end, some applications are given.

In this paper some new general fractional integral inequalities for convex and m-convex functions by involving an extended Mittag-Leffler function are presented. These results produce inequalities for several kinds of fractional integral operators. Some interesting special cases of our main results are also pointed out.

In this paper we show how the Shannon entropy is connected to the theory of majorization. They are both linked to the measure of disorder in a system. However, the theory of majorization usually gives stronger criteria than the entropic inequalities. We give some generalized results for majorization inequality using Csiszár f-divergence. This divergence, applied to some special convex functions, reduces the results for majorization inequality in the form of Shannon entropy and the Kullback-Leibler divergence. We give several applications by using the Zipf-Mandelbrot law.

Abstract. We give generalization of majorization theorem for the class of n -convex functions by using Taylor's formula and Green function. We use inequalities for theČebyšev functional to obtain bounds for the identities related to generalizations of majorization inequalities. We present mean value theorems and n -exponential convexity for the functional obtained from the generalized majorization inequalities. At the end we discuss the results for particular families of function and give means.Mathematics subject classification (2010): 26D15, 26D20, 26D99.

Generalized results of majorization inequality are obtained by using newly Green functions defined in [N. Mahmood, R. P. Agarwal, S. I. Butt, J. Pečarić, J. Inequal. Appl., 2017 (2017), 17 pages] and Lidstone's polynomial. We find new upper bounds of Grüss and Ostrowski type. We give further results of majorization inequality by making linear functionals constructed on convex functionsx . Some applications are given.