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.