一般地，假设 a 1 , a 2 , ⋯ , a n {\displaystyle a_{1},a_{2},\cdots ,a_{n}} 为n个非负实数,它们的算术平均值记为
A n = a 1 + a 2 + ⋯ + a n n {\displaystyle \mathbf {A} _{n}={\frac {a_{1}+a_{2}+\cdots +a_{n}}{n}}}
几何平均值记为
G n = a 1 ⋅ a 2 ⋯ a n n {\displaystyle \mathbf {G} _{n}={\sqrt[{n}]{a_{1}\cdot a_{2}\cdots a_{n}}}}
则
A n ⩾ G n {\displaystyle A_{n}\geqslant G_{n}}
当且仅 a 1 = a 2 = ⋯ = a n {\displaystyle a_{1}=a_{2}=\cdots =a_{n}} 时,等号成立.
上述不等式即为平均值不等式，简称均值不等式
为n个非负实数,它们的算术平均值记为
![{\displaystyle \mathbf {G} _{n}={\sqrt[{n}]{a_{1}\cdot a_{2}\cdots a_{n}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9c9a16dcb65dd4915a726b9d6b0951add5c7384)
时,等号成立.
