A complement to Diananda's inequality

Peng Gao

doi:10.7153/mia-2018-21-19

A complement to Diananda's inequality

Peng Gao^*

^*此作品的通讯作者

数学科学学院

科研成果: 期刊稿件 › 文章 › 同行评审

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摘要

Let Mn,r =(Σⁿ_i=1 qix^r_i)1r, r=0 and Mn,0 =limr→0Mn,r be the weighted power means of n non-negative numbers xi with qi > 0 satisfying Σⁿ_i=1 qi = 1. In particular, An = Mn,1 , Gn = Mn,0 are the arithmetic and geometric means of these numbers, respectively. A result of Diananda shows that Mn,1/2 -qAn -(1-q)Gn ≥ 0, Mn,1/2 -(1-q)An -qGn ≤ 0, where q = minqi . In this paper, we prove analogue inequalities in the reversed direction.

源语言	英语
页（从-至）	251-263
页数	13
期刊	Mathematical Inequalities and Applications
卷	21
期	1
DOI	https://doi.org/10.7153/mia-2018-21-19
出版状态	已出版 - 1月 2018

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N2 - Let Mn,r =(Σni=1 qixri)1r, r=0 and Mn,0 =limr→0Mn,r be the weighted power means of n non-negative numbers xi with qi > 0 satisfying Σni=1 qi = 1. In particular, An = Mn,1 , Gn = Mn,0 are the arithmetic and geometric means of these numbers, respectively. A result of Diananda shows that Mn,1/2 -qAn -(1-q)Gn ≥ 0, Mn,1/2 -(1-q)An -qGn ≤ 0, where q = minqi . In this paper, we prove analogue inequalities in the reversed direction.

AB - Let Mn,r =(Σni=1 qixri)1r, r=0 and Mn,0 =limr→0Mn,r be the weighted power means of n non-negative numbers xi with qi > 0 satisfying Σni=1 qi = 1. In particular, An = Mn,1 , Gn = Mn,0 are the arithmetic and geometric means of these numbers, respectively. A result of Diananda shows that Mn,1/2 -qAn -(1-q)Gn ≥ 0, Mn,1/2 -(1-q)An -qGn ≤ 0, where q = minqi . In this paper, we prove analogue inequalities in the reversed direction.

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JO - Mathematical Inequalities and Applications

JF - Mathematical Inequalities and Applications

IS - 1

ER -

A complement to Diananda's inequality

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