正数和与积的对偶不等式

Dual inequalities for sum and product of positive numbers

给出了在个数相同的两组正数之积相等的情形下,判定这两组正数之和大小的一个充分条件,由此得出了正数和最大值存在的链式条件．应用这一链式条件解决了将,nl个正数分成等个数的n个数组后,每组之积的和与每组之和的积的最值问题．对于正数积的最小值问题,证明了与正数和最大值类似的结果,且相应的定理之间互为对偶定理．

For two finite sets of n positive numbers with the same products of n numbers in each set, we give and prove a sufficient condition of determining which sum is small or large in the two sums of n numbers in each set. Thereby, we derive a chain condition of existing maximum values for the sum of all numbers in the set. After partitioning a set of nl positive numbers into n subsets with l numbers, the extreme values of the sum of the n the products of l numbers in each subset and of the product of the n the sums of l numbers in each subset are obtained easily by using the chain condition. For the minimum value problem of the product of positive numbers, we prove the similar results to the maximum value problem of the sum of positive numbers, and the corresponding theorems are dual ones each other.