• An Optimal Version of an Inequality Involving the Third Symmetric Means

• Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/118/04/0505-0516

• Keywords

Third symmetric mean of 𝑘 degree; optimal values; inequality; descending dimension method.

• Abstract

Let $(GA)^{[k]}_n(a), A_n(a), G_n(a)$ be the third symmetric mean of 𝑘 degree, the arithmetic and geometric means of $a_1,\ldots,a_n(a_i&gt;0,i=1,\ldots,n)$, respectively. By means of descending dimension method, we prove that the maximum of 𝑝 is $\frac{k-1}{n-1}$ and the minimum of 𝑞 is $\frac{n}{n-1}\left(\frac{k-1}{k}\right)^{\frac{k}{n}}$ so that the inequalities

$$(G_n(a))^{1-p}(A_n(a))^p\leq (GA)^{[k]}_n(a)\leq (1-q)G_n(a)+q A_n(a) (2\leq k\leq n-1)$$

hold.

• Author Affiliations

1. Department of Mathematics and Computer Science, Chengdu University, 610106 Chengdu, China
2. School of Mathematics and Computer Science, Nanjing Normal University, 210097 Nanjing, China
3. Department of Mathematics, Shaoxing University Shangyu College, 312300 Zhejiang, China

• Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 3
June 2019

• Proceedings – Mathematical Sciences | News

© 2017-2019 Indian Academy of Sciences, Bengaluru.