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.

This paper is devoted to obtain generalized results related to majorization-type inequalities by using well-known Fink's identity and new types of Green functions, introduced by Mehmood et al. (J. Inequal. Appl. 2017:108, 2017). We give a generalized majorization theorem for the class of n-convex functions. We utilize the Csiszár f-divergence and generalized majorization-type inequalities in providing the corresponding generalizations. As an application, we present the obtained results in terms of Shannon entropy and Kullback-Leibler distance.

This paper begins with a rigorous study of convex functions with the goal of developing the majorization theorems in the form of Taylor representation. In this paper, some new types of Green functions, introduced by Pečarić-Agarwal-Butt-Mehmood (2017) [11] and Taylor's formula, are used to obtain the identities related to majorization type inequalities. We present the monotonicity of the linear functionals deduced from our generalized results by using the family of (n + 1) -convex functions at a point. We give upper bounds and mean value theorems for obtained generalized identities. At the end, we explore some applications. (2010): 26A51, 26D15, 26D20, 26D99. Mathematics subject classification

In this paper we give generalized results of a majorization inequality by using extension of the Montgomery identity and newly defined Green's functions (Mehmood et al. in J. Inequal. Appl. 2017(1):108, 2017). We obtain a generalized majorization theorem for the class of n-convex functions. We use Csiszár f-divergence and generalized majorization-type inequalities to obtain new generalized results. We further discuss our obtained generalized results in terms of the Shannon entropy and the Kullback-Leibler distance.

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