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

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      Permanent link:
      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 $a_1,\ldots,a_n(a_i>0,i=1,\ldots,n)$, 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

       

      Wen Jiajin1 Yuan Jun2 Yuan Shufeng3

      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
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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