【摘要】 用降维法建立了含n个正实数a1,a2,…,an的第一k次对称平均∑kn(a)=nnk-1∑1≤i1<…<ik≤n∏j=1aij1/k,第二k次对称平均σnk(a)=kn-11≤i1<∑…<ik≤n(ai1ai2…aik)1/k,第三k次对称平均∏nk(a)=∏1≤i1<…<ik≤nai1+ai2+…+aikknk-1的一个不等式链∏k(a)≥∑n+1-k(a)≥σnn+1-k(a)(1<k<n),并将此结果用于正定矩阵及单形.更多还原

【Abstract】 In this paper,the authors established a chain of inequalities involving first symmertic mean of k degree ∑^k_n（a）=nk^-1∑1≤i₁<…<i_k≤n∏kj=1a_ij^1/k,and second symmetric mean of k degree σ^k_n（a）=nk^-1∑1≤i₁<…<i_k≤n(a_i1a_i2…a_ik)^1/k and third symmetric mean of k degree ∏^k_n（a）=∏1≤i₁<…<i_k≤na_i1+a_i2+…+a_ikk^nk-1 as follows.∏^k_n（a）≥∑^n+1-k_n（a）≥σ^n+1-k_n（a）,where a∶=（a₁,…,a_n）,a_i>0（i=1,…,n）,1<k<n.Then,these results are applied to positive definite matrice and simplexes.更多还原