幂平均不等式的最优值  

On the Optimal Values for Inequalities Involving Power Means

作　　者：王挽澜[1] 文家金[1] 石焕南 

机构地区：[1]成都大学数学与计算机科学系,成都610106 [2]北京联合大学电气工程系,北京100011

出　　处：《数学学报（中文版）》2004年第6期1053-1062,共10页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金资助项目(10171073);成都大学科研基金资助项目

摘　　要：设Mn[r](a)为a的r阶幂平均,0<α<θ<β,那么满足不等式[Mn[α](a)]1-λ.[Mn[β](a)]λ≤Mn[θ](a)的最大实数λ是λ≥{1+(β-θ)/[m(θ-α)]}-1.这里m=min{[2+(n-2)tβ]/[2+(n-2)tα],t∈R++};满足反向不等式的最小实数λ是λ=[β(θ-α)]/[θ(β-α)].本文的方法基于优势理论与解析技巧,对于建立不等式的最优化思想作了尽可能多的展示.作为应用,得到了一些涉及和、积分与矩阵的新不等式(含Hardy不等式的推广与加强).Let Mn[r] (a) be the r-th power mean of a, O <α<θ<β. Then the largest number λ, satisfying [Mn[α](a)]1-λ[Mn[β](a)]λ≤Mn[θ](a), is λ≥{1+ (β-θ)/[m(θ-α)]}-1, where m = min{[2 + (n - 2)tβ]/[2 + (n - 2)tα], t∈ R++}; the smallest number λ, satisfying the reverse inequality, is λ= [β(θ-α)]/[θ(β-α)]. The method depends on the theory of majorization and analytic techniques. The optimality idea of establish-ing inequalities is displayed as many times as possible. As some applications, several inequalities (including improved Hardy's inequality) involving sums, integrals and ma-trices are obtained.

关 键 词：不等式 幂平均 最优值 

分 类 号：O178[理学—数学]