对称平均对幂平均的分隔及其应用

Separation of Power Means by Symmetric Mean

设 ∑kn(Xn)是n个正实数x1,… ,xn(n≥ 3)的k( 2 ≤k≤n- 1 )次对称平均 ,而Mt(Xn)为x1,… ,xn 的t次幂平均 .本文获得了使不等式Mp(Xn) ≤ ∑kn(Xn) ≤Mq(Xn)成立的 p的最大值和q的最小值 ,其中k=2 ,… ,n- 1 ,并将此结果用于n维长方体及文 [2 ]的征解问题 6 1 .

Let ∑kn(X n) be k th(2≤ k≤n- 1) symmetric mean for n positive reals x 1,…,x n(n ≥3),and M t(X n) be the power mean for x 1,…,x n. In this paper we obtain the maximum of p and the minmum of q such that M p(X n)≤∑kn(X n)≤M q(X n) hold, where k=2,3,…,n-1. As some applications of this result, we also obtain another interesting results for n -cuboid and open problem 61 in [2].