抽象受控不等式的同构映射

The Isomorphic Mapping of the Abstract Majorization Inequalities

通过提出抽象平均、抽象凸函数、抽象控制和抽象受控不等式的同构映射概念,建立了抽象凸函数同构映射的基本定理:设(■)_F和(■)_S为抽象平均,α(x)为严格单调(■)_(F-)-函数,β(x)为严格单调递增(■)_(S-)-函数,那么f(x)为抽象(■)_F→(■)_S严格上凸函数的充分必要条件是:f*(x)=β^(-1)o f oα(x)为抽象(■)_F~α→(■)_S~β严格上凸函数,这里(■)_F~α=α^(-1)o(■)oα,(■)_S~β=β^(-1)o(■)_S oβ.在抽象平均同构映射的基础上,获得了抽象受控不等式同构映射的基本定理:记a_i=α^(-1)(x_i),b_i=α^(-1)(yi)(i=1,2,…,n),则不等式(■)_S{f(x_1),f(x_2),…,f(x_n)}>(■)_S{f(y_1),f(y_2),…,f(y_n)}成立的充分必要条件是:不等式(■)_S~β{f~*(a_1),f~*(a_2),…,f~*(a_n)}>(■)_S~β{f~*(b_1),f~*(b_2),…,f~*(b_n)}成立.作为基本定理的简单应用,证明了算术受控不等式、几何受控不等式和调和受控不等式这三类不等式是同构的.简而言之,这三类受控不等式是等价的.

Using the axiomatic method,the isomorphic mapping of abstract mean,abstract convex function,abstract majorization and abstract majorization inequality are proposed,respectively.The fundamental theorems about the isomorphic mapping of abstract convex functions are established as follows:Suppose that(■)_F and(■)_S are abstract means,α(x) is a strict monotone(■)_F-function,β(x) is a strict monotone increasing(■)_S-function,then the f(x) is abstract strict(■)_F →(■)_S convex function if and only if f~*(x) = β^(-1) o f o α(x) is abstract strict()_F~α→(■)_S~β convex function,where(■)_F~α = α^(-1) o(■) o α,(■)_S~β = β^(-1) o(■)_S o β.The fundamental theorems about isomorphic mapping of abstract majorization inequalities are established as follows:let a_i = α^(-1)(x_i),b_i=α^(-1)(y_i)(i = 1,2,…,n),then the inequalities(■)_S{f(x_1),f(x_2),…,f(x_2),…,f(x_n)} >(■)_S{f(y_1),f(y_2),…,f(y_n)}hold if and only if the inequalities(■)_S~β{f~*(a_1),f~*(a_2),…,f~*(a_n)}>(■)_S~β{f~*(b_1),f~*(b_2),…,f~*(b_n)}.hold.As their applications,we prove that the arithmetic majorization inequalities,geometric majorization inequalities and harmonic majorization inequalities are isomorphic,i.e.,three classes of majorization inequalities are equivalent.