幂平均不等式的最优值

设Mnr](a)为a的r阶幂平均,0<α<θ<β,那么满足不等式Mnα](a)]1-λ.Mnβ](a)]λ≤Mnθ](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).

On the Optimal Values for Inequalities Involving Power Means

Let Mnr] (a) be the r-th power mean of a, O <α<θ<β. Then the largest number λ, satisfying Mnα](a)]1-λMnβ](a)]λ≤Mnθ](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.