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,resp...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 meanΣandΣ,and abstractΣ→Σstrict convex function f(x)on the interval I,if xi,yi∈I(i=1,2,...,n)satisfy that(x1,x2,...,xn)<nΣ(y1,y2,...,yn),thenΣ{f(x1),f(x2),...,f(xn)}≥Σ{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 SRn,and n-variable abstract symmetricalΣ→Σstrict convex functionφ()on S,if,■∈S satisfynΣ■,thenφ(x)〈(y);if vector groupxi,yi∈S(i=1,2,...,m)satisfy{x1,x2,...,xm}〈Σn{y1,y2,...,ym},thenΣ{φ(x1),φ(x2),...,φ(xm)}Σ{φ(y1),φ(y2),...,φ(ym)}.展开更多
基金supported by the National Key Basic Research Project of China(Grant No.2004CB318003)the Foundation of the Education Department of Sichuan Province of China(Grant No.07ZA087)
文摘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 meanΣandΣ,and abstractΣ→Σstrict convex function f(x)on the interval I,if xi,yi∈I(i=1,2,...,n)satisfy that(x1,x2,...,xn)<nΣ(y1,y2,...,yn),thenΣ{f(x1),f(x2),...,f(xn)}≥Σ{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 SRn,and n-variable abstract symmetricalΣ→Σstrict convex functionφ()on S,if,■∈S satisfynΣ■,thenφ(x)〈(y);if vector groupxi,yi∈S(i=1,2,...,m)satisfy{x1,x2,...,xm}〈Σn{y1,y2,...,ym},thenΣ{φ(x1),φ(x2),...,φ(xm)}Σ{φ(y1),φ(y2),...,φ(ym)}.
