Majorization And Its Applications In Elementary Mathematics

It is well known that inequalities have to play an important role in nearly all branches of mathematics. It is an important method to prove symmetric inequalities by using Jensen Inequality. But the requirement of this method is very strong: the function must be convex or concave. Whereas many functions we can not judge their convexity, especially for multi‐functions, so we can't use Jensen's method, but the conditions when the functions get the maximum or minimum is the same as using Jensen's method. Urged by this necessity, we try to find ways to promote the Jensen's method to prove these inequalities uniformly. Using majorization and Schur‐convex function just can solve this problem. Since the majorization itself is defined by a series of inequalities, we can also use the majorizations and their properties to prove inequalities.This thesis introduces the majorization methods in detail to prove or extend inequalities which in elementary mathematics, especially in Mathematics Olympiad. The first chapter The Basic Definitions and Theorems of Majorization is the basis theories of majorization. It focuses on the properties of majorization and its wide ranges of colorful equivalent conditions. The second chapter The Basic Definitions and Theorems of Schur‐convexity introduces the applications of majorization in convex functions and Schur‐convex functions. The third chapter lists some beautiful conclusions. Such as some sufficient or necessary conditions for Schur‐convexity of a function of two variables F(x,y) = [f (y)‐ f (x)] / [g(y)‐ g (x)], and the theorems of pseudo‐elementary symmetric functions similarly in sharp of F(x,y,z) = f (x,y) + f(y,z) + f (z,x) and their n variables forms. These results are applied to yield some new inequalities in a triangle. The last chapter discusses the applications of majorization in elementary mathematics. Firstly we talk about the applications in trigonometric inequalities. Because this method can be formulated for a function in any number of variables, with ternary function to pave the way, we will soon use this method into algebraic inequalities. And from the methodological perspective, we discuss the trigonometric inequalities in the strong majorization. As a supplementary of the method, we will discuss the inequalities about the edge element of a triangle and algebraic inequalities in the weak majorization. Just like the majorization can be extended formulaically in n dimensions, we can extend the Permutation Inequality to n pairs of arrays for deriving WEIWEI Dual Inequality.Taking use of majorizations in elementary mathematics aims to show a feature of majorization methods that we can use a uniform method to prove a large number of known inequalities easily, but also to derive or discover new inequalities incidentally. This thesis also fully embodies the characteristics of majorization. We use its theorems to derive some brand new inequalities (see 4.80‐4.99), which are not appeared before.