平均數不等式維基百科,自由的 encyclopedia 平均數不等式,或稱平均值不等式、均值不等式,是數學上的一組不等式,也是算術-幾何平均值不等式的推廣。它是說: x 1 , x 2 , … , x n ∈ R + ⇒ n ∑ i = 1 n 1 x i ≤ ∏ i = 1 n x i n ≤ ∑ i = 1 n x i n ≤ ∑ i = 1 n x i 2 n {\displaystyle x_{1},x_{2},\ldots ,x_{n}\in \mathbb {R_{+}} \Rightarrow {\dfrac {n}{\displaystyle \sum _{i=1}^{n}{\dfrac {1}{x_{i}}}}}\leq {\sqrt[{n}]{\prod _{i=1}^{n}x_{i}}}\leq {\dfrac {\displaystyle \sum _{i=1}^{n}x_{i}}{n}}\leq {\sqrt {\dfrac {\displaystyle \sum _{i=1}^{n}x_{i}^{2}}{n}}}} 即 H n ≤ G n ≤ A n ≤ Q n {\displaystyle H_{n}\leq G_{n}\leq A_{n}\leq Q_{n}} 其中: H n = n ∑ i = 1 n 1 x i = n 1 x 1 + 1 x 2 + ⋯ + 1 x n {\displaystyle H_{n}={\dfrac {n}{\displaystyle \sum _{i=1}^{n}{\dfrac {1}{x_{i}}}}}={\dfrac {n}{{\dfrac {1}{x_{1}}}+{\dfrac {1}{x_{2}}}+\cdots +{\dfrac {1}{x_{n}}}}}} G n = ∏ i = 1 n x i n = x 1 x 2 ⋯ x n n {\displaystyle G_{n}={\sqrt[{n}]{\prod _{i=1}^{n}x_{i}}}={\sqrt[{n}]{x_{1}x_{2}\cdots x_{n}}}} A n = ∑ i = 1 n x i n = x 1 + x 2 + ⋯ + x n n {\displaystyle A_{n}={\dfrac {\displaystyle \sum _{i=1}^{n}x_{i}}{n}}={\dfrac {x_{1}+x_{2}+\cdots +x_{n}}{n}}} Q n = ∑ i = 1 n x i 2 n = x 1 2 + x 2 2 + ⋯ + x n 2 n {\displaystyle Q_{n}={\sqrt {\dfrac {\displaystyle \sum _{i=1}^{n}x_{i}^{2}}{n}}}={\sqrt {\dfrac {x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}}{n}}}} 若且唯若 x 1 = x 2 = ⋯ = x n {\displaystyle x_{1}=x_{2}=\cdots =x_{n}} ,等號成立。 即對這些正實數:調和平均數 ≤ 幾何平均數 ≤ 算術平均數 ≤ 平方平均數(方均根) 簡記為:「調幾算方」
平均數不等式,或稱平均值不等式、均值不等式,是數學上的一組不等式,也是算術-幾何平均值不等式的推廣。它是說: x 1 , x 2 , … , x n ∈ R + ⇒ n ∑ i = 1 n 1 x i ≤ ∏ i = 1 n x i n ≤ ∑ i = 1 n x i n ≤ ∑ i = 1 n x i 2 n {\displaystyle x_{1},x_{2},\ldots ,x_{n}\in \mathbb {R_{+}} \Rightarrow {\dfrac {n}{\displaystyle \sum _{i=1}^{n}{\dfrac {1}{x_{i}}}}}\leq {\sqrt[{n}]{\prod _{i=1}^{n}x_{i}}}\leq {\dfrac {\displaystyle \sum _{i=1}^{n}x_{i}}{n}}\leq {\sqrt {\dfrac {\displaystyle \sum _{i=1}^{n}x_{i}^{2}}{n}}}} 即 H n ≤ G n ≤ A n ≤ Q n {\displaystyle H_{n}\leq G_{n}\leq A_{n}\leq Q_{n}} 其中: H n = n ∑ i = 1 n 1 x i = n 1 x 1 + 1 x 2 + ⋯ + 1 x n {\displaystyle H_{n}={\dfrac {n}{\displaystyle \sum _{i=1}^{n}{\dfrac {1}{x_{i}}}}}={\dfrac {n}{{\dfrac {1}{x_{1}}}+{\dfrac {1}{x_{2}}}+\cdots +{\dfrac {1}{x_{n}}}}}} G n = ∏ i = 1 n x i n = x 1 x 2 ⋯ x n n {\displaystyle G_{n}={\sqrt[{n}]{\prod _{i=1}^{n}x_{i}}}={\sqrt[{n}]{x_{1}x_{2}\cdots x_{n}}}} A n = ∑ i = 1 n x i n = x 1 + x 2 + ⋯ + x n n {\displaystyle A_{n}={\dfrac {\displaystyle \sum _{i=1}^{n}x_{i}}{n}}={\dfrac {x_{1}+x_{2}+\cdots +x_{n}}{n}}} Q n = ∑ i = 1 n x i 2 n = x 1 2 + x 2 2 + ⋯ + x n 2 n {\displaystyle Q_{n}={\sqrt {\dfrac {\displaystyle \sum _{i=1}^{n}x_{i}^{2}}{n}}}={\sqrt {\dfrac {x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}}{n}}}} 若且唯若 x 1 = x 2 = ⋯ = x n {\displaystyle x_{1}=x_{2}=\cdots =x_{n}} ,等號成立。 即對這些正實數:調和平均數 ≤ 幾何平均數 ≤ 算術平均數 ≤ 平方平均數(方均根) 簡記為:「調幾算方」