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重要不等式

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=…=an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=…=an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=…=an时成立.

(4)当α<0<β< i="">  时,对正数ai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=…=an时成立.

柯西不等式

ai, bi  (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

ai,bi,…,li(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.

ai,bi(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

ai,bi>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

ai>0,bi>0 (i=1,2,…,n).若a1a2≤…≤an,且b1b2≤…≤bn,或a1a2≥…≥an,且b1b2≥…≥bn,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b1b2≤…≤bna1a2≥…≥an,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

ai>0(i=1,2,…,n),且0 <rs,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

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 \[ b^{\frac{1}{n}} - 1 < \frac{{b - 1}}{n} \]