A refinement of the arithmetic mean-geometric mean inequality
A refinement of the arithmetic mean-geometric mean inequality
复制标题
DOI：
10.1090/s0002-9939-1978-0476971-2
复制
发表时间：
1978
影响因子：
2.9
通讯作者：
D. I. Cartwright;M. Field
中科院分区：
文献类型：
--
作者：
D. I. Cartwright;M. Field
给出了n个正实数的算术平均值与几何平均值之差的上下界。在这篇注记中，我们证明了算术平均-几何平均不等式的一个简单的精化。我们的结果解决了Kenneth S.Williams在[5]中提出的一个问题，推广了[3]第215页上的一个不等式。平均值之间差异的其他估计在[2]、[3]和[4]中讨论。定理。设xk E[a，b]和pk>；0，其中a>；0，其中a>；0，且假设，-lpk=1.然后，记x=E4=ipkxk，有2pk(Xkx)2xi(Xkp)0，且设yt=fb t dm(T)且a2 f=b(t-_[)2 dm(T)为m的均值和方差，则2b a2<；exp(log(T)dm(T))2<；A2这源于我们的定理和在[a，b]上的所有概率测度的集合中的形式为en lPk6kx(其中6x表示集中在点x的概率测度)的弱*密度。(见[1，第709页]。)请注意，编辑们于1977年8月15日收到的不等式/b expt log(T)dm(T)<；tt。AMS(MOS)主题分类(1970)。小学26A87。
Upper and lower bounds are given for the difference between the arithmetic and geometric means of n positive real numbers in terms of the variance of these numbers. In this note we prove a simple refinement of the arithmetic mean-geometric mean inequality. Our result solves a problem posed by Kenneth S. Williams in [5] and generalizes an inequality on p. 215 of [3]. Other estimates for the difference between the means are discussed in [2], [3] and [4]. THEOREM. Suppose that Xk E [a, b] and Pk > 0 for k = 1, ... , n, where a > 0, and suppose that ,-lPk = 1. Then, writing x = E4=ipkxk, we have 2 Pk(Xk x)2X I (Xkp) 0, and let yt = fb t dm(t) and a2 f= b(t -_ [)2 dm(t) be the mean and variance of m. Then 2b a2 < exp(( log(t) dm(t)) 2< a2 This follows from our theorem and the weak* density of the measures of the form En lPk6kx (where 6x denotes the probability measure which is concentrated at the point x) in the set of all probability measures on [a, b]. (See [1, p. 709].) Notice that the inequality /b expt log(t) dm (t) < tt Received by the editors August 15, 1977. AMS (MOS) subject classifications (1970). Primary 26A87.