The fundamental theory of abstract majorization inequalities

Abstract

Using the axiomatic method, abstract concepts such as abstract mean, abstract convex function and abstract majorization are proposed. They are the generalizations of concepts of mean, convex function and majorization, respectively. Through the logical deduction, the fundamental theorems about abstract majorization inequalities are established as follows: for arbitrary abstract mean Σ and \( \Sigma ' \) and abstract ∑ ↪ \( \Sigma ' \) strict convex function f(x) on the interval I, if xi, yi ∈ I (i = 1, 2,..., n) satisfy that \( (x_1 ,x_2 , \ldots ,x_n ) \prec _n^\Sigma (y_1 ,y_2 , \ldots ,y_n ) \) then \( \Sigma ' \){f(x1), f(x2),..., f(xn)} ⩾ \( \Sigma ' \){f(y1), f(y2),..., f(yn)}. This class of inequalities extends and generalizes the fundamental theorem of majorization inequalities. Moreover, concepts such as abstract vector mean are proposed, the fundamental theorems about abstract majorization inequalities are generalized to n-dimensional vector space. The fundamental theorem of majorization inequalities about the abstract vector mean are established as follows: for arbitrary symmetrical convex set \( \mathcal{S} \subset \mathbb{R}^n \), and n-variable abstract symmetrical \( \overline \Sigma \) ↪ \( \Sigma ' \) strict convex function \( \phi (\bar x) \) on \( \mathcal{S} \), if \( \bar x,\bar y \in \mathcal{S} \), satisfy \( \bar x \prec _n^\Sigma \bar y \), then \( \phi (\bar x) \geqslant \phi (\bar y) \); if vector group \( \bar x_i ,\bar y_i \in \mathcal{S}(i = 1,2, \ldots ,m) \) satisfy \( \{ \bar x_1 ,\bar x_2 , \ldots ,\bar x_m \} \prec _n^{\bar \Sigma } \{ \bar y_1 ,\bar y_2 , \ldots ,\bar y_m \} \), then \( \Sigma '\{ \phi (\bar x_1 ),\phi (\bar x_2 ), \ldots ,\phi (\bar x_m )\} \geqslant \Sigma '\{ \phi (\bar y_1 ),\phi (\bar y_2 ), \ldots ,\phi (\bar y_m )\} \).