摘要
设∏kn(a)=(∏1 i1<…<ik≤nk-1∑kj=1aij)1(nk),A(a)=1n∑ni=1ai,G(a)=(∏ni=1ai)1/n分别为n个正实数a1,…,an的第三k次对称平均,算术平均,几何平均 本文用降维法建立了如下的一个优化不等式:若a∈Rn++,n≥3,则使不等式[A(a)]p[G(a)]1-p≤∏n-1n(a)≤qA(a)+(1-q)G(a)成立的实数p的最大值是pn=n-2n-1,实数q的最小值是qn=n(n-2)1-1n(n-1)1n-2。
Let ∏~k_n(a)=(∏1i_1<...<i_k≤nD)k^(-1)∑~k_(j=1)a_(i_j))^(1/(nk)),A(a)=1n∑~n_(i=1)a_i,G(a)=(∏~n_(i=1)a_i)^(1/n)are the third symmetric mean of k dgree, the arithmatic mean and the geometric mean of a_1,...,a_n. By means of decsending dimension method, we have proved that the maximun of p is p_n=n-2n-1and the minimum of q is q_n=n(n-2)^(1-1n)(n-1)^(1n-2)so that the inequality ~p^(1-p)≤∏^(n-1)_n(a)≤q A(a)+(1-q)G(a) is held, where, a∈R^n_(++),n≥3. As applications, our results are applied to positive difinite matrice and simplexes.
出处
《成都大学学报（自然科学版）》
2004年第3期1-10,共10页
Journal of Chengdu University（Natural Science Edition）
