A CHAIN OF EIGHT INEQUALITIES INVOLVING MEANS OF TWO ARGUMENTS

被引:0
|
作者
Mestrovic, Romeo [1 ]
机构
[1] Univ Montenegro, Maritime Fac Kotor, Dobrota 85330, Kotor, Montenegro
来源
TEACHING OF MATHEMATICS | 2024年 / 27卷 / 01期
关键词
Harmonic mean; geometric mean; arithmetic mean; quadratic mean; H-G-A-Q inequality; Muirhead's inequality;
D O I
10.57016/TM-XDVI2817
中图分类号
G40 [教育学];
学科分类号
040101 ; 120403 ;
摘要
For two positive real numbers a and b, let H: = H(a, b ), G:= G(a, b), A := A(a, b ) and Q : Q(a, b) be the harmonic mean, the geometric mean, the arithmetic mean and the quadratic mean of a and b, respectively. In this short note, we prove the following interesting chain involving eight inequalities: G <= root QH <= root AG <= A+G/2 <= Q + H/2 <= root A(2) + G(2)/2 <= root Q(2) + H-2/2 <= Q + G/2 <= A, where equality holds in each of these inequalities if and only if a = b . Some remarks, in particular connected with Muirhead's inequality, and two questions related to a similar form of chain of inequalities are also given.
引用
收藏
页码:27 / 32
页数:6
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