Majorization and its Applications on Some Functions

This paper reviews the special case of an order which is called Majorization ordering. It generalizes vector Majorization and some applications that have come after the publication of Marshall and Olkin Inequalities. It presents the basic properties of Majorization and two important kinds of Majorization which are Weakly Supermajorization and Weakly Submajorization and some relations between them. Furthermore, this paper also contains maps from R^n to R^m which preserve various orders that most of these orders are elementary and useful characterizations of Majorization, as Majorization together with the strongly related concept of Schur-convexity gives an important characterization of convex functions that expresses preservation of order rather than convexity. Also in this study, examples are used to explore the characteristics of majorization, weakly supermajorization, and weakly submajorization as well as the relationships between them. We described the application of majorization on various functions, such as monotonic functions, convex functions, and so on, with some properties by taking into account the concept of our title majorization and its applications on some Functions. Theorems and examples are used to explain such outcomes.