+ + + =

算术平均值与几何平均值不等式

（1）几个数的算术平均值的绝对值不超过这些数的均方根，即 \[ \left| {\frac{{a_1 + a_2 + \cdots + a_n }}{n}} \right| \le \sqrt {\frac{{a_1 ^2 + a_2 ^2 + \cdots + a_n ^2 }}{n}} \] 等号只当a₁=a₂=…=a_n时成立．

（2）几个正数的几何平均值不超过这些数的算术平均值，即 \[ \sqrt[n]{{a_1 a_2 \cdots a_n }} \le \frac{{a_1 + a_2 + \cdots + a_n }}{n} \] 等号只当a₁=a₂=…=a_n时成立．

（3）对几个正数的加权几何平均值有 \[ a_1 ^{p_1 } a_2 ^{p_2 } \cdots a_n ^{p_n } \le \left( {\frac{{p_1 a_1 + p_2 a_2 + \cdots + p_n a_n }}{{p_1 + p_2 + p_3 + \cdots + p_n }}} \right)^{p_1 + p_2 + p_3 + \cdots + p_n } \] 等号只当a₁=a₂=…=a_n时成立．

（4）当α<0<β< i="">  时，对正数a_i有 \[ \left( {\frac{1}{n}\sum\limits_{i = 1}^n {a_i ^\alpha } } \right)^{\frac{1}{\alpha }} \le \left( {a_1 a_2 \cdots a_n } \right)^{\frac{1}{n}} \le \left( {\frac{1}{n}\sum\limits_{i = 1}^n {a_i ^\beta } } \right)^{\frac{1}{\beta }} \] 等号只当a₁=a₂=…=a_n时成立．

柯西不等式

设a_i, b_i  (i=1,2,…,n)为任意实数，则 \[ \left( {\sum\limits_{i = 1}^n {a_i b_i } } \right)^2 \le \left( {\sum\limits_{i = 1}^n {a_i ^2 } } \right)\left( {\sum\limits_{i = 1}^n {b_i ^2 } } \right) \] 等号只当 \[ \frac{{a_1 }}{{b_1 }} = \frac{{a_2 }}{{b_2 }} = \cdots = \frac{{a_n }}{{b_n }} \] 时成立．这个不等式表明一个角（取实数值）的余弦值总是小于1的，或者说二矢量内积小于二矢量长度之积．

赫尔德不等式

设a_i,b_i,…,l_i(i=1,2,…,n)为正数，又α,β,…,λ为正数，且α+β+…+λ=1，则 \[ \sum\limits_{i = 1}^n {a_i ^\alpha b_i ^\beta } \cdots l_i ^\lambda \le \left( {\sum\limits_{i = 1}^n {a_i } } \right)^\alpha \left( {\sum\limits_{i = 1}^n {b_i } } \right)^\beta \cdots \left( {\sum\limits_{i = 1}^n {l_i } } \right)^\lambda \] 等号只当 \[ \frac{{a_k }}{{\sum {a_i } }} = \frac{{b_k }}{{\sum {b_i } }} = \cdots = \frac{{l_k }}{{\sum {l_i } }} \] 时成立．

设a_i,b_i(i=1,2,…,n)为正数，又k>0,k≠1,k' 和k共轭，即 \[ \frac{1}{{k^1 }} + \frac{1}{k} = 1或\left( {k - 1} \right)\left( {k' - 1} \right) = 1 \] 则 \[ \begin{array}{l} \sum\limits_{i = 1}^n {a_i b_i } \le \left( {\sum\limits_{i = 1}^n {a_i ^k } } \right)^{\frac{1}{k}} \left( {\sum\limits_{i = 1}^n {b_i ^{k'} } } \right)^{\frac{1}{{k'}}} \left( {k > 1} \right) \\ \sum\limits_{i = 1}^n {a_i b_i } \le \left( {\sum\limits_{i = 1}^n {a_i ^k } } \right)^{\frac{1}{k}} \left( {\sum\limits_{i = 1}^n {b_i ^{k'} } } \right)^{\frac{1}{{k'}}} \left( {k < 1} \right) \\ \end{array} \] 等号只当\[ \frac{{a_1 }}{{b_1 }} = \frac{{a_2 }}{{b_2 }} = \cdots = \frac{{a_n }}{{b_n }} \] 时成立

闵可夫斯基不等式

设a_i,b_i>0 (i=1,2,…,n)，又r>0,r≠1,则 \[ \begin{array}{l} \left\{ {\sum\limits_{i = 1}^n {\left( {a_i + b_i } \right)^r } } \right\}^{\frac{1}{r}} \le \left( {\sum\limits_{i = 1}^n {a_i ^r } } \right)^{\frac{1}{r}} + \left( {\sum\limits_{i = 1}^n {b_i ^r } } \right)^{\frac{1}{r}} \left( {r > 1} \right) \\ \left\{ {\sum\limits_{i = 1}^n {\left( {a_i + b_i } \right)^r } } \right\}^{\frac{1}{r}} \, \ge \left( {\sum\limits_{i = 1}^n {a_i ^r } } \right)^{\frac{1}{r}} + \left( {\sum\limits_{i = 1}^n {b_i ^r } } \right)^{\frac{1}{r}} \left( {r < 1} \right) \\ \end{array} \] 等号只当\[ \frac{{a_1 }}{{b_1 }} = \frac{{a_2 }}{{b_2 }} = \cdots = \frac{{a_n }}{{b_n }} \] 时成立，当r=2时，此不等式也称为三角形不等式，它表明三角形两边之和大于第三边．

契贝谢夫不等式

设a_i>0,b_i>0 (i=1,2,…,n)．若a₁≤a₂≤…≤a_n，且b₁≤b₂≤…≤b_n，或a₁≥a₂≥…≥a_n，且b₁≥b₂≥…≥b_n，则 \[ \left( {\frac{1}{n}\sum\limits_{i = 1}^n {a_i } } \right)\left( {\frac{1}{n}\sum\limits_{i = 1}^n {b_i } } \right) \le \frac{1}{n}\sum\limits_{i = 1}^n {a_i b_i } \] 若b₁≤b₂≤…≤b_n而a₁≥a₂≥…≥a_n，则 \[ \left( {\frac{1}{n}\sum\limits_{i = 1}^n {a_i } } \right)\left( {\frac{1}{n}\sum\limits_{i = 1}^n {b_i } } \right) \ge \frac{1}{n}\sum\limits_{i = 1}^n {a_i b_i } \]

詹生不等式

设a_i>0(i=1,2,…,n)，且0 <r≤s，则\[ \left( {\sum\limits_{i = 1}^n {a_i ^s } } \right)^{\frac{1}{s}} \le \left( {\sum\limits_{i = 1}^n {a_i ^r } } \right)^{\frac{1}{r}} \]

伯努利不等式

设a>1，自然数n>1，则 \[ a^n > 1 + n\left( {a - 1} \right) \] 特别令 \[ a = b^{\frac{1}{n}} \left( {b > 1} \right) \] 则 \[ b^{\frac{1}{n}} - 1 < \frac{{b - 1}}{n} \]