• Yuan Jun

Articles written in Proceedings – Mathematical Sciences

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

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.

• # Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 5
November 2019

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Posted on July 25, 2019