Holder means

In the mathematics is generalized mean , the Hölder average (by Otto Holder , 1859-1937) or the power means (engl. U. A. (p-th) power mean ) a (sometimes of ) a generalized average . The term is not uniform, names like - te agents , means the order or the degree or with exponent are also in circulation. In English it is also known as a generalized mean . ${\displaystyle p}$ ${\displaystyle p}$

The spellings are just as inconsistent, instead of being written , or . ${\displaystyle H_{p}}$ ${\displaystyle M_{p}(x)}$ ${\displaystyle m_{p}(x)}$ ${\displaystyle \mu _{p}(x)}$

The Hölder mean generalizes the mean values ​​known since the Pythagoreans such as the arithmetic , geometric , quadratic and harmonic mean by introducing a parameter ${\displaystyle p.}$

definition

For a real number , the Hölder mean of the numbers for the level is defined as ${\displaystyle p\neq 0}$ ${\displaystyle x_{1},\ldots ,x_{n}\geq 0}$ ${\displaystyle p}$

${\displaystyle M_{p}(x_{1},\dots ,x_{n})=\left({\frac {1}{n}}\cdot \sum _{i=1}^{n}x_{i}^{p}\right)^{1/p}={\sqrt[{p}]{\frac {x_{1}^{p}+x_{2}^{p}+\ldots +x_{n}^{p}}{n}}}}$,

where the root notation is usually only used for natural numbers . ${\displaystyle p}$

A suitable definition for is ${\displaystyle p=0}$

${\displaystyle M_{0}(x_{1},\ldots ,x_{n}):=\lim _{s\to 0}M_{s}(x_{1},\ldots ,x_{n}).}$

properties

• The Holder mean is homogeneous in terms of , that is ${\displaystyle x_{1}\ldots ,x_{n}}$

${\displaystyle M_{p}(\alpha \,x_{1},\ldots ,\alpha \,x_{n})=\alpha \cdot M_{p}(x_{1},\ldots ,x_{n})}$

• Also applies

${\displaystyle M_{p}(x_{1},\dots ,x_{n\cdot k})=M_{p}(M_{p}(x_{1},\dots ,x_{k}),M_{p}(x_{k+1},\dots ,x_{2\cdot k}),\dots ,M_{p}(x_{(n-1)\cdot k+1},\dots ,x_{n\cdot k}))}$

• An important inequality to the Hölder means is

${\displaystyle p<q\quad \Rightarrow \quad M_{p}(x_{1},\ldots ,x_{n})\leq M_{q}(x_{1},\ldots ,x_{n})}$

From this follows (special cases) the inequality of the mean values

${\displaystyle \min(x_{1},\ldots ,x_{n})\leq {\bar {x}}_{\mathrm {harm} }\leq {\bar {x}}_{\mathrm {geom} }\leq {\bar {x}}_{\mathrm {arithm} }\leq {\bar {x}}_{\mathrm {quadr} }\leq {\bar {x}}_{\mathrm {kubisch} }\leq \max(x_{1},\ldots ,x_{n})}$

• The power means are quite simply related to the sample moments around zero: ${\displaystyle m_{p}}$

${\displaystyle {\bar {x}}(p)={\sqrt[{p}]{m_{p}}}}$

• In stochastics , the convergence in the p-th mean is defined using these power mean values.

Special cases

Graphic representation of different mean values ​​of two values a, b

The known mean values are obtained by selecting a suitable parameter : ${\displaystyle p}$

	${\displaystyle \lim _{p\to -\infty }}$	${\displaystyle M_{p}(x_{1},\dots ,x_{n})}$	${\displaystyle =\min\{x_{1},\dots ,x_{n}\}}$	minimum
${\displaystyle p=-1}$		${\displaystyle M_{-1}(x_{1},\dots ,x_{n})}$	${\displaystyle ={\frac {n}{{\frac {1}{x_{1}}}+\dots +{\frac {1}{x_{n}}}}}}$	Harmonious mean
	${\displaystyle \lim _{p\to 0}}$	${\displaystyle M_{p}(x_{1},\dots ,x_{n})}$	${\displaystyle ={\sqrt[{n}]{x_{1}\cdot \dots \cdot x_{n}}}}$	Geometric mean
${\displaystyle p=1}$		${\displaystyle M_{1}(x_{1},\dots ,x_{n})}$	${\displaystyle ={\frac {x_{1}+\dots +x_{n}}{n}}}$	Arithmetic mean
${\displaystyle p=2}$		${\displaystyle M_{2}(x_{1},\dots ,x_{n})}$	${\displaystyle ={\sqrt {\frac {x_{1}^{2}+\dots +x_{n}^{2}}{n}}}}$	Square mean
${\displaystyle p=3}$		${\displaystyle M_{3}(x_{1},\dots ,x_{n})}$	${\displaystyle ={\sqrt[{3}]{\frac {x_{1}^{3}+\dots +x_{n}^{3}}{n}}}}$	Cubic mean
	${\displaystyle \lim _{p\to \infty }}$	${\displaystyle M_{p}(x_{1},\dots ,x_{n})}$	${\displaystyle =\max\{x_{1},\dots ,x_{n}\}}$	maximum

Further generalizations

Weighted Holder mean

Also to the generalized mean a can weighted mean define: The weighted generalized mean can be with the weights with defined as ${\displaystyle \omega _{1},\omega _{2},\ldots ,\omega _{n}}$ ${\displaystyle \omega _{1}+\omega _{2}+\ldots +\omega _{n}=1}$

${\displaystyle {M_{\omega }}^{p}=\left(\omega _{1}\cdot x_{1}^{p}+\omega _{2}\cdot x_{2}^{p}+\ldots +\omega _{n}\cdot x_{n}^{p}\right)^{1/p},}$

where the unweighted Hölder mean is used. ${\displaystyle \omega _{1}=\omega _{2}=\ldots =\omega _{n}={\tfrac {1}{n}}}$

f-means

Compare quasi-arithmetic mean

The Holder means can be further generalized

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

or weighted to

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

Where is a function of ; the Holder means used . ${\displaystyle f}$ ${\displaystyle x}$ ${\displaystyle \,f(x)=x^{p}}$

More examples :

• If the returns on an investment are in the years to , the average return is obtained as the mean of the individual returns for the function . ${\displaystyle x_{1},\ldots ,x_{n}\geq 0}$ ${\displaystyle 1}$ ${\displaystyle n}$ ${\displaystyle f}$ ${\displaystyle \,f(x)=\ln(1+x)}$
• If the ages of persons are, the actuarial average age is obtained as the mean of the individual ages for the function ; where means the death intensity. In practice, the sum-weighted actuarial average age is relevant; here the ages of the insured persons are weighted with the respective insured sums; the death intensity is often replaced by the one year probability of death . ${\displaystyle x_{1},\ldots ,x_{n}}$ ${\displaystyle n}$ ${\displaystyle f}$ ${\displaystyle \,f(x)=\mu _{x}}$ ${\displaystyle \,\mu _{x}}$ ${\displaystyle \,q_{x}}$

See also

• Clay means
• Average
• Stolarsky mean , logarithmic mean

literature

• Julian Havil : Gamma: Euler's constant, prime number beaches and the Riemann hypothesis , Springer, Berlin 2007, ISBN 978-3-540-48495-0
• PS Bulls: Handbook of Means and Their Inequalities . Dordrecht, Netherlands: Kluwer, 2003, pp. 175–265

Web links

• Eric W. Weisstein : Power mean . In: MathWorld (English).
• Weighted Power Mean and Proof on planetmath.org
• Examples of Generalized Mean
• Jutta's math newsletter