不等式Sum(1/a_i)(from i=1 to i=n)≥n^2/Sum(a_i)(from i=1 to i=n)的两个推广

Two teneralizations of inequalitySum(1/a_i)(from i=1 to i=n)≥n^2/Sum(a_i)(from i=1 to i=n)

从n个正数的调和平均值 ,不大于其算术平均值出发 ,导出了一个分式不等式 ,又应用排序的方法 ,从两个方面将这个不等式加以推广 ,得到了两个更具一般形式的分式不等式和指数不等式 :∑ni=1biai ≥n∑ni =1 bi∑ni=1 ai,∑ni =1 aim≥(∑ni=1 ai) mnm - 1 (n∈N ,m∈Z)。

From a hasmonic average consisting of n ferms of positive integral which is less than or equal to its arityemetic average,a fractional inquality is derived.Then by using sequential method and generalization of the inquality with two aspects,we can also get two general types of fractional inquality and exponential inquality:∑ni=1 b ia i≥n∑ni=1b i∑ni=1a i,∑ni=1a i m≥(∑ni=1a i) mn m-1 (n∈N,m∈Z).