A complement to Diananda's inequality

Peng Gao

doi:10.7153/mia-2018-21-19

A complement to Diananda's inequality

Peng Gao^*

^*Corresponding author for this work

School of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

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.

Original language	English
Pages (from-to)	251-263
Number of pages	13
Journal	Mathematical Inequalities and Applications
Volume	21
Issue number	1
DOIs	https://doi.org/10.7153/mia-2018-21-19
State	Published - Jan 2018

Keywords

Power means

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title = "A complement to Diananda's inequality",

abstract = "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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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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ER -

A complement to Diananda's inequality

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