An optimal version of an inequality involving the third symmetric means (Q732830)

Let \(a_i>0\) \((i= 1,2,\dots, n)\) and denote by \((GA)^{[k]}_n\), \(A_n\), \(G_n\) the third symmetric mean of degree \(k\), resp. the arithmetic and geometric means of these numbers. The authors prove the inequalities \[ (G_n)^{1-p}(A_n)^p\leq (GA)^{[k]}_n\leq (1- q)G_n+ qA_n\quad\text{for }2\leq k\leq n-1, \] with the best values of \(p\) and \(q\) being \(p= (k-1)/(n-1)\), \(q= {n\over n-1} ({k-1\over 2})^{k/n}\). In the proof, the so-called method of descending dimension is applied [see e.g. \textit{J. J. Wen} et al., J. Xinan Nat. Univ. (Nat. Sci. ed.) 29, No. 5, 527--532 (2003)].