# 幂平均

（重定向自赫尔德平均

## 定义

${\displaystyle p}$  是一非零实数，可定义实数 ${\displaystyle x_{1},\dots ,x_{n}}$  p次幂平均

${\displaystyle M_{p}(x_{1},\dots ,x_{n})=\left({\frac {1}{n}}\cdot \sum _{i=1}^{n}x_{i}^{p}\right)^{\frac {1}{p}}\,}$

## 性质

• 和所有平均一样，幂平均是各参数 ${\displaystyle x_{1},\dots ,x_{n}}$  的一次齐次函数。即若 ${\displaystyle b}$  是一个正实数，则 ${\displaystyle b\cdot x_{1},\dots ,b\cdot x_{n}}$  指数为 ${\displaystyle p}$  的幂平均等于 ${\displaystyle b}$ ${\displaystyle x_{1},\dots ,x_{n}}$  的幂平均。
• 几何算术平均一样，这种平均的计算可以分解成同样大小的子块来计算。
${\displaystyle M_{p}(x_{1},\dots ,x_{n\cdot k})=M_{p}(M_{p}(x_{1},\dots ,x_{k}),M_{p}(x_{k+1},\dots ,x_{2\cdot k}),\dots ,M_{p}(x_{(n-1)\cdot k+1},\dots ,x_{n\cdot k}))}$

### 幂平均不等式

${\displaystyle \forall p\in \mathbb {R} \ {\frac {\partial M_{p}(x_{1},\dots ,x_{n})}{\partial p}}\geq 0,}$

## 特例

 ${\displaystyle M_{-\infty }(x_{1},\dots ,x_{n})=\lim _{p\to -\infty }M_{p}(x_{1},\dots ,x_{n})=\min\{x_{1},\dots ,x_{n}\}}$ 最小值 ${\displaystyle M_{-1}(x_{1},\dots ,x_{n})={\frac {n}{{\frac {1}{x_{1}}}+\dots +{\frac {1}{x_{n}}}}}}$ 调和平均 ${\displaystyle M_{0}(x_{1},\dots ,x_{n})=\lim _{p\to 0}M_{p}(x_{1},\dots ,x_{n})={\sqrt[{n}]{x_{1}\cdot \dots \cdot x_{n}}}}$ 几何平均 ${\displaystyle M_{1}(x_{1},\dots ,x_{n})={\frac {x_{1}+\dots +x_{n}}{n}}}$ 算术平均 ${\displaystyle M_{2}(x_{1},\dots ,x_{n})={\sqrt {\frac {x_{1}^{2}+\dots +x_{n}^{2}}{n}}}}$ 平方平均 ${\displaystyle M_{3}(x_{1},\dots ,x_{n})={\sqrt[{3}]{\frac {x_{1}^{3}+\dots +x_{n}^{3}}{n}}}}$ 立方平均 ${\displaystyle M_{+\infty }(x_{1},\dots ,x_{n})=\lim _{p\to \infty }M_{p}(x_{1},\dots ,x_{n})=\max\{x_{1},\dots ,x_{n}\}}$ 最大值

## 幂平均不等式的证明

### 不同符号的不等式之等价

${\displaystyle {\sqrt[{p}]{\sum _{i=1}^{n}w_{i}x_{i}^{p}}}\leq {\sqrt[{q}]{\sum _{i=1}^{n}w_{i}x_{i}^{q}}}}$

${\displaystyle {\sqrt[{p}]{\sum _{i=1}^{n}{\frac {w_{i}}{x_{i}^{p}}}}}\leq {\sqrt[{q}]{\sum _{i=1}^{n}{\frac {w_{i}}{x_{i}^{q}}}}}}$ .

${\displaystyle {\sqrt[{-p}]{\sum _{i=1}^{n}w_{i}x_{i}^{-p}}}={\sqrt[{p}]{\frac {1}{\sum _{i=1}^{n}w_{i}{\frac {1}{x_{i}^{p}}}}}}\geq {\sqrt[{q}]{\frac {1}{\sum _{i=1}^{n}w_{i}{\frac {1}{x_{i}^{q}}}}}}={\sqrt[{-q}]{\sum _{i=1}^{n}w_{i}x_{i}^{-q}}}}$ ,

### 几何平均

${\displaystyle \prod _{i=1}^{n}x_{i}^{w_{i}}\leq {\sqrt[{q}]{\sum _{i=1}^{n}w_{i}x_{i}^{q}}}}$
${\displaystyle {\sqrt[{q}]{\sum _{i=1}^{n}w_{i}x_{i}^{q}}}\leq \prod _{i=1}^{n}x_{i}^{w_{i}}}$

（第一个不等式对正数 q，第二个对负数）

${\displaystyle \prod _{i=1}^{n}x_{i}^{w_{i}\cdot q}\leq \sum _{i=1}^{n}w_{i}x_{i}^{q}}$

${\displaystyle \sum _{i=1}^{n}w_{i}\log(x_{i})\leq \log \left(\sum _{i=1}^{n}w_{i}x_{i}\right)}$
${\displaystyle \log \left(\prod _{i=1}^{n}x_{i}^{w_{i}}\right)\leq \log \left(\sum _{i=1}^{n}w_{i}x_{i}\right)}$

${\displaystyle \prod _{i=1}^{n}x_{i}^{w_{i}}\leq \sum _{i=1}^{n}w_{i}x_{i}.}$

${\displaystyle {\sqrt[{-q}]{\sum _{i=1}^{n}w_{i}x_{i}^{-q}}}\leq \prod _{i=1}^{n}x_{i}^{w_{i}}\leq {\sqrt[{q}]{\sum _{i=1}^{n}w_{i}x_{i}^{q}}}.}$

${\displaystyle \lim _{q\rightarrow 0}{\sqrt[{q}]{\sum _{i=1}^{n}w_{i}x_{i}^{q}}}=\prod _{i=1}^{n}x_{i}^{w_{i}}}$

### 幂平均不等式

${\displaystyle {\sqrt[{p}]{\sum _{i=1}^{n}w_{i}x_{i}^{p}}}\leq {\sqrt[{q}]{\sum _{i=1}^{n}w_{i}x_{i}^{q}}}.}$

${\displaystyle {\sqrt[{p}]{\sum _{i=1}^{n}w_{i}x_{i}^{p}}}\leq \prod _{i=1}^{n}x_{i}^{w_{i}}\leq {\sqrt[{q}]{\sum _{i=1}^{n}w_{i}x_{i}^{q}}}}$

${\displaystyle f(\sum _{i=1}^{n}w_{i}x_{i}^{p})\leq \sum _{i=1}^{n}w_{i}f(x_{i}^{p})}$
${\displaystyle {\sqrt[{\frac {p}{q}}]{\sum _{i=1}^{n}w_{i}x_{i}^{p}}}\leq \sum _{i=1}^{n}w_{i}x_{i}^{q}}$

${\displaystyle {\sqrt[{p}]{\sum _{i=1}^{n}w_{i}x_{i}^{p}}}\leq {\sqrt[{q}]{\sum _{i=1}^{n}w_{i}x_{i}^{q}}}.}$

## 最小值与最大值

${\displaystyle \min(x_{1},x_{2},\ldots ,x_{n})\leq {\sqrt[{q}]{\sum _{i=1}^{n}w_{i}x_{i}^{q}}}\leq \max(x_{1},x_{2},\ldots ,x_{n})}$

${\displaystyle {\sqrt[{q}]{\sum _{i=1}^{n}w_{i}x_{i}^{q}}}\leq x_{1}}$

${\displaystyle \sum _{i=1}^{n}w_{i}x_{i}^{q}\leq {\color {red}\geq }x_{1}^{q}}$

q>0 为 ≤，q<0 为 ≥

${\displaystyle \sum _{i=2}^{n}w_{i}x_{i}^{q}\leq {\color {red}\geq }(1-w_{1})x_{1}^{q}}$

${\displaystyle \sum _{i=2}^{n}{\frac {w_{i}}{(1-w_{1})}}x_{i}^{q}\leq {\color {red}\geq }x_{1}^{q}}$

1 - w1 不为零，从而：

${\displaystyle \sum _{i=2}^{n}{\frac {w_{i}}{(1-w_{1})}}=1}$

${\displaystyle \sum _{i=2}^{n}{\frac {w_{i}}{(1-w_{1})}}(x_{i}^{q}-x_{1}^{q})\leq {\color {red}\geq }0}$

${\displaystyle x_{i}^{q}-x_{1}^{q}\leq {\color {red}\geq }0}$

${\displaystyle {\sqrt[{q}]{w_{1}x_{1}^{q}}}<{\sqrt[{q}]{\sum _{i=1}^{n}w_{i}x_{i}^{q}}}\leq x_{1},}$

${\displaystyle q}$  趋于 ${\displaystyle +\infty }$  时，左边同样趋于 ${\displaystyle x_{1}}$ ，由夹逼定理知中间项幂平均趋于 ${\displaystyle x_{1}}$ 。最小值的证明完全类似。

## 广义 f-平均

${\displaystyle M_{f}(x_{1},\dots ,x_{n})=f^{-1}\left[{{\frac {1}{n}}\cdot \sum _{i=1}^{n}{f(x_{i})}}\right]}$

## 应用

### 信号处理

 powerSmooth :: Floating a => ([a] -> [a]) -> a -> [a] -> [a]
powerSmooth smooth p =
map (** recip p) . smooth . map (**p)

• 对於大 ${\displaystyle p}$  值，这可作为一个整流信号的包封檢測器envelope detector）。
• 对於小 ${\displaystyle p}$  值，这可作为一个質量譜基线侦测器baseline detector）。