n个非负实数a_1,…,a_n的算术平均数与几何平均数之间有这样的关系: (a_1+…+a_n)/n≥(a_1·…·a_n.)~(1/2) (1)其中“=”当且仅当a_1,=…=a_n时成立。这就是著名的算术——几何平均值定理。这个定理的证法很多,在此就不再赘述了。本文主要介绍算术——几何平均值定理的一个推广图式,以及它在证明不等式中的应用。为便于叙述,我们记 There is a relationship between the arithmetic mean of the n non-negative real numbers a_1,...,a_n and the geometric mean: (a_1+...+a_n)/n≥(a_1····a_n.)~(1/2) (1 ) where “=” is true if and only if a_1, =... = a_n. This is the famous arithmetic-geometric mean theorem. The proof of this theorem is many and it will not be repeated here. This paper mainly introduces a generalized graph of arithmetic-geometric mean theorem and its application in proving inequalities. For ease of description we remember