An optimal version of an inequality involving the third symmetric means

Let (GA) 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 {fx505-1} hold.