An optimal version of an inequality involving the third symmetric means

作者：Wen Jiajin; Yuan Jun; Yuan Shufeng

来源：Proceedings of the Indian Academy of Sciences - Mathematical Sciences, 2008, 118(4): 505-516.

DOI：10.1007/s12044-008-0038-0

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

Let (G A)(n)([k])(a), A(n)(a), G(n)(a) be the third symmetric mean of k degree, the arithmetic and geometric means of a(1), ... , a(n) (a(i) > 0, i = 1, ... , n), respectively. By means of descending dimension method, we prove that the maximum of p is k-1/n-1 and the minimum of q is n/n-1(k-1/k)(k/n) so that the inequalities
(G(n)(a))(1-p)(A(n)(a))(p) <= (GA)(n)([k])(a) <= (1 - q)G(n) (a) + qA(n) (a) (2 <= k <= n - 1)
hold.

• 出版日期2008-11
• 单位南京师范大学; 成都大学