# Generalized mean

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N-th root of the arithmetic mean of the given numbers raised to the power n

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Plot of several generalized means

          M

            p

        (
        1
        ,
        x
        )

    {\displaystyle M_{p}(1,x)}

In [mathematics](/source/Mathematics), **generalized means** (or **power mean** or **Hölder mean** from [Otto Hölder](/source/Otto_H%C3%B6lder))[1] are a family of functions for aggregating sets of numbers. These include as special cases the [Pythagorean means](/source/Pythagorean_means) ([arithmetic](/source/Arithmetic_mean), [geometric](/source/Geometric_mean), and [harmonic](/source/Harmonic_mean) [means](/source/Mean)).

## Definition

If p is a non-zero [real number](/source/Real_number), and x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} are [positive real numbers](/source/Positive_real_numbers), then the **generalized mean** or **power mean** with exponent p of these positive real numbers is[2][3]

M p ( x 1 , … , x n ) = ( 1 n ∑ i = 1 n x i p ) 1 / p . {\displaystyle M_{p}(x_{1},\dots ,x_{n})=\left({\frac {1}{n}}\sum _{i=1}^{n}x_{i}^{p}\right)^{{1}/{p}}.}

(See [p-norm](/source/Norm_(mathematics)#p-norm)). For *p* = 0 we set it equal to the geometric mean (which is the limit of means with exponents approaching zero, as proved below):

M 0 ( x 1 , … , x n ) = ( ∏ i = 1 n x i ) 1 / n . {\displaystyle M_{0}(x_{1},\dots ,x_{n})=\left(\prod _{i=1}^{n}x_{i}\right)^{1/n}.}

Furthermore, for a [sequence](/source/Sequence) of positive weights wi we define the **weighted power mean** as[2] M p ( x 1 , … , x n ) = ( ∑ i = 1 n w i x i p ∑ i = 1 n w i ) 1 / p {\displaystyle M_{p}(x_{1},\dots ,x_{n})=\left({\frac {\sum _{i=1}^{n}w_{i}x_{i}^{p}}{\sum _{i=1}^{n}w_{i}}}\right)^{{1}/{p}}} and when *p* = 0, it is equal to the [weighted geometric mean](/source/Weighted_geometric_mean):

M 0 ( x 1 , … , x n ) = ( ∏ i = 1 n x i w i ) 1 / ∑ i = 1 n w i . {\displaystyle M_{0}(x_{1},\dots ,x_{n})=\left(\prod _{i=1}^{n}x_{i}^{w_{i}}\right)^{1/\sum _{i=1}^{n}w_{i}}.}

The unweighted means correspond to setting all *wi* = 1.

## Special cases

For some values of p {\displaystyle p} , the mean M p ( x 1 , … , x n ) {\displaystyle M_{p}(x_{1},\dots ,x_{n})} corresponds to a well known mean.

A visual depiction of some of the specified cases for

        n
        =
        2

    {\displaystyle n=2}

.
  Harmonic mean:

          M

            −
            1

        (
        a
        ,
        b
        )

    {\displaystyle M_{-1}(a,b)}

.

  Geometric mean:

          M

            0

        (
        a
        ,
        b
        )

    {\displaystyle M_{0}(a,b)}

.

  Arithmetic mean:

          M

            1

        (
        a
        ,
        b
        )

    {\displaystyle M_{1}(a,b)}

.

  Quadratic mean:

          M

            2

        (
        a
        ,
        b
        )

    {\displaystyle M_{2}(a,b)}

.

Name Exponent Value Minimum p = − ∞ {\displaystyle p=-\infty } min { x 1 , … , x n } {\displaystyle \min\{x_{1},\dots ,x_{n}\}} Harmonic mean p = − 1 {\displaystyle p=-1} n 1 x 1 + ⋯ + 1 x n {\displaystyle {\frac {n}{{\frac {1}{x_{1}}}+\dots +{\frac {1}{x_{n}}}}}} Geometric mean p = 0 {\displaystyle p=0} x 1 … x n n {\displaystyle {\sqrt[{n}]{x_{1}\dots x_{n}}}} Arithmetic mean p = 1 {\displaystyle p=1} x 1 + ⋯ + x n n {\displaystyle {\frac {x_{1}+\dots +x_{n}}{n}}} Root mean square p = 2 {\displaystyle p=2} x 1 2 + ⋯ + x n 2 n {\displaystyle {\sqrt {\frac {x_{1}^{2}+\dots +x_{n}^{2}}{n}}}} Cubic mean p = 3 {\displaystyle p=3} x 1 3 + ⋯ + x n 3 n 3 {\displaystyle {\sqrt[{3}]{\frac {x_{1}^{3}+\dots +x_{n}^{3}}{n}}}} Maximum p = + ∞ {\displaystyle p=+\infty } max { x 1 , … , x n } {\displaystyle \max\{x_{1},\dots ,x_{n}\}}

**Proof of lim p → 0 M p = M 0 {\textstyle \lim _{p\to 0}M_{p}=M_{0}} (geometric mean)**

For the purpose of the proof, we will assume without loss of generality that w i ∈ [ 0 , 1 ] {\displaystyle w_{i}\in [0,1]} and ∑ i = 1 n w i = 1. {\displaystyle \sum _{i=1}^{n}w_{i}=1.}

We can rewrite the definition of M p {\displaystyle M_{p}} using the exponential function as

M p ( x 1 , … , x n ) = exp ⁡ ( ln ⁡ [ ( ∑ i = 1 n w i x i p ) 1 / p ] ) = exp ⁡ ( ln ⁡ ( ∑ i = 1 n w i x i p ) p ) {\displaystyle M_{p}(x_{1},\dots ,x_{n})=\exp {\left(\ln {\left[\left(\sum _{i=1}^{n}w_{i}x_{i}^{p}\right)^{1/p}\right]}\right)}=\exp {\left({\frac {\ln {\left(\sum _{i=1}^{n}w_{i}x_{i}^{p}\right)}}{p}}\right)}}

In the limit *p* → 0, we can apply [L'Hôpital's rule](/source/L'H%C3%B4pital's_rule) to the argument of the exponential function. We assume that p ∈ R {\displaystyle p\in \mathbb {R} } but *p* ≠ 0, and that the sum of wi is equal to 1 (without loss in generality);[4] differentiating the numerator and denominator with respect to p, we have lim p → 0 ln ⁡ ( ∑ i = 1 n w i x i p ) p = lim p → 0 ∑ i = 1 n w i x i p ln ⁡ x i ∑ j = 1 n w j x j p 1 = lim p → 0 ∑ i = 1 n w i x i p ln ⁡ x i ∑ j = 1 n w j x j p = ∑ i = 1 n w i ln ⁡ x i ∑ j = 1 n w j = ∑ i = 1 n w i ln ⁡ x i = ln ⁡ ( ∏ i = 1 n x i w i ) {\displaystyle {\begin{aligned}\lim _{p\to 0}{\frac {\ln {\left(\sum _{i=1}^{n}w_{i}x_{i}^{p}\right)}}{p}}&=\lim _{p\to 0}{\frac {\frac {\sum _{i=1}^{n}w_{i}x_{i}^{p}\ln {x_{i}}}{\sum _{j=1}^{n}w_{j}x_{j}^{p}}}{1}}\\&=\lim _{p\to 0}{\frac {\sum _{i=1}^{n}w_{i}x_{i}^{p}\ln {x_{i}}}{\sum _{j=1}^{n}w_{j}x_{j}^{p}}}\\&={\frac {\sum _{i=1}^{n}w_{i}\ln {x_{i}}}{\sum _{j=1}^{n}w_{j}}}\\&=\sum _{i=1}^{n}w_{i}\ln {x_{i}}\\&=\ln {\left(\prod _{i=1}^{n}x_{i}^{w_{i}}\right)}\end{aligned}}}

By the continuity of the exponential function, we can substitute back into the above relation to obtain lim p → 0 M p ( x 1 , … , x n ) = exp ⁡ ( ln ⁡ ( ∏ i = 1 n x i w i ) ) = ∏ i = 1 n x i w i = M 0 ( x 1 , … , x n ) {\displaystyle \lim _{p\to 0}M_{p}(x_{1},\dots ,x_{n})=\exp {\left(\ln {\left(\prod _{i=1}^{n}x_{i}^{w_{i}}\right)}\right)}=\prod _{i=1}^{n}x_{i}^{w_{i}}=M_{0}(x_{1},\dots ,x_{n})} as desired.[2]

**Proof of lim p → ∞ M p = M ∞ {\textstyle \lim _{p\to \infty }M_{p}=M_{\infty }} and lim p → − ∞ M p = M − ∞ {\textstyle \lim _{p\to -\infty }M_{p}=M_{-\infty }}**

Assume (possibly after relabeling and combining terms together) that x 1 ≥ ⋯ ≥ x n {\displaystyle x_{1}\geq \dots \geq x_{n}} . Then

lim p → ∞ M p ( x 1 , … , x n ) = lim p → ∞ ( ∑ i = 1 n w i x i p ) 1 / p = x 1 lim p → ∞ ( ∑ i = 1 n w i ( x i x 1 ) p ) 1 / p = x 1 = M ∞ ( x 1 , … , x n ) . {\displaystyle {\begin{aligned}\lim _{p\to \infty }M_{p}(x_{1},\dots ,x_{n})&=\lim _{p\to \infty }\left(\sum _{i=1}^{n}w_{i}x_{i}^{p}\right)^{1/p}\\&=x_{1}\lim _{p\to \infty }\left(\sum _{i=1}^{n}w_{i}\left({\frac {x_{i}}{x_{1}}}\right)^{p}\right)^{1/p}\\&=x_{1}=M_{\infty }(x_{1},\dots ,x_{n}).\end{aligned}}}

The formula for M − ∞ {\displaystyle M_{-\infty }} follows from M − ∞ ( x 1 , … , x n ) = 1 M ∞ ( 1 / x 1 , … , 1 / x n ) = x n . {\displaystyle M_{-\infty }(x_{1},\dots ,x_{n})={\frac {1}{M_{\infty }(1/x_{1},\dots ,1/x_{n})}}=x_{n}.}

## Properties

Let x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} be a sequence of positive real numbers, then the following properties hold:[1]

1. min ( x 1 , … , x n ) ≤ M p ( x 1 , … , x n ) ≤ max ( x 1 , … , x n ) {\displaystyle \min(x_{1},\dots ,x_{n})\leq M_{p}(x_{1},\dots ,x_{n})\leq \max(x_{1},\dots ,x_{n})} . Each generalized mean always lies between the smallest and largest of the x values.

1. M p ( x 1 , … , x n ) = M p ( P ( x 1 , … , x n ) ) {\displaystyle M_{p}(x_{1},\dots ,x_{n})=M_{p}(P(x_{1},\dots ,x_{n}))} , where P {\displaystyle P} is a permutation operator. Each generalized mean is a symmetric function of its arguments; permuting the arguments of a generalized mean does not change its value.

1. M p ( b x 1 , … , b x n ) = b ⋅ M p ( x 1 , … , x n ) {\displaystyle M_{p}(bx_{1},\dots ,bx_{n})=b\cdot M_{p}(x_{1},\dots ,x_{n})} . Like most [means](/source/Mean#Properties), the generalized mean is a [homogeneous function](/source/Homogeneous_function) of its arguments *x*1, ..., *xn*. That is, if b is a positive real number, then the generalized mean with exponent p of the numbers b ⋅ x 1 , … , b ⋅ x n {\displaystyle b\cdot x_{1},\dots ,b\cdot x_{n}} is equal to b times the generalized mean of the numbers *x*1, ..., *xn*.

1. M p ( x 1 , … , x n ⋅ k ) = M p [ M p ( x 1 , … , x k ) , M p ( x k + 1 , … , x 2 ⋅ k ) , … , M p ( x ( n − 1 ) ⋅ k + 1 , … , x n ⋅ k ) ] {\displaystyle M_{p}(x_{1},\dots ,x_{n\cdot k})=M_{p}\left[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})\right]} . Like the [quasi-arithmetic means](/source/Quasi-arithmetic_mean), the computation of the mean can be split into computations of equal sized sub-blocks. This enables use of a [divide and conquer algorithm](/source/Divide_and_conquer_algorithm) to calculate the means, when desirable.

### Generalized mean inequality

Geometric [proof without words](/source/Proof_without_words) that *max* (*a*,*b*) > [root mean square](/source/Root_mean_square) (**RMS**) or [quadratic mean](/source/Quadratic_mean) (**QM**) > [arithmetic mean](/source/Arithmetic_mean) (**AM**) > [geometric mean](/source/Geometric_mean) (**GM**) > [harmonic mean](/source/Harmonic_mean) (**HM**) > *min* (*a*,*b*) of two distinct positive numbers *a* and *b*[note 1]

In general, if *p* < *q*, then M p ( x 1 , … , x n ) ≤ M q ( x 1 , … , x n ) {\displaystyle M_{p}(x_{1},\dots ,x_{n})\leq M_{q}(x_{1},\dots ,x_{n})} and the two means are equal if and only if *x*1 = *x*2 = ... = *xn*.

The inequality is true for real values of p and q, as well as positive and negative infinity values.

It follows from the fact that, for all real p, ∂ ∂ p M p ( x 1 , … , x n ) ≥ 0 {\displaystyle {\frac {\partial }{\partial p}}M_{p}(x_{1},\dots ,x_{n})\geq 0} which can be proved using [Jensen's inequality](/source/Jensen's_inequality).

In particular, for p in {−1, 0, 1}, the generalized mean inequality implies the [Pythagorean means](/source/Pythagorean_means) inequality as well as the [inequality of arithmetic and geometric means](/source/Inequality_of_arithmetic_and_geometric_means).

## Proof of the weighted inequality

We will prove the weighted power mean inequality. For the purpose of the proof we will assume the following [without loss of generality](/source/Without_loss_of_generality): w i ∈ [ 0 , 1 ] ∑ i = 1 n w i = 1 {\displaystyle {\begin{aligned}w_{i}\in [0,1]\\\sum _{i=1}^{n}w_{i}=1\end{aligned}}}

The proof for unweighted power means can be easily obtained by substituting *wi* = 1/*n*.

### Equivalence of inequalities between means of opposite signs

Suppose an average between power means with exponents p and q holds: ( ∑ i = 1 n w i x i p ) 1 / p ≥ ( ∑ i = 1 n w i x i q ) 1 / q {\displaystyle \left(\sum _{i=1}^{n}w_{i}x_{i}^{p}\right)^{1/p}\geq \left(\sum _{i=1}^{n}w_{i}x_{i}^{q}\right)^{1/q}} applying this, then: ( ∑ i = 1 n w i x i p ) 1 / p ≥ ( ∑ i = 1 n w i x i q ) 1 / q {\displaystyle \left(\sum _{i=1}^{n}{\frac {w_{i}}{x_{i}^{p}}}\right)^{1/p}\geq \left(\sum _{i=1}^{n}{\frac {w_{i}}{x_{i}^{q}}}\right)^{1/q}}

We raise both sides to the power of −1 (strictly decreasing function in positive reals): ( ∑ i = 1 n w i x i − p ) − 1 / p = ( 1 ∑ i = 1 n w i 1 x i p ) 1 / p ≤ ( 1 ∑ i = 1 n w i 1 x i q ) 1 / q = ( ∑ i = 1 n w i x i − q ) − 1 / q {\displaystyle \left(\sum _{i=1}^{n}w_{i}x_{i}^{-p}\right)^{-1/p}=\left({\frac {1}{\sum _{i=1}^{n}w_{i}{\frac {1}{x_{i}^{p}}}}}\right)^{1/p}\leq \left({\frac {1}{\sum _{i=1}^{n}w_{i}{\frac {1}{x_{i}^{q}}}}}\right)^{1/q}=\left(\sum _{i=1}^{n}w_{i}x_{i}^{-q}\right)^{-1/q}}

We get the inequality for means with exponents −*p* and −*q*, and we can use the same reasoning backwards, thus proving the inequalities to be equivalent, which will be used in some of the later proofs.

### Geometric mean

For any *q* > 0 and non-negative weights summing to 1, the following inequality holds: ( ∑ i = 1 n w i x i − q ) − 1 / q ≤ ∏ i = 1 n x i w i ≤ ( ∑ i = 1 n w i x i q ) 1 / q . {\displaystyle \left(\sum _{i=1}^{n}w_{i}x_{i}^{-q}\right)^{-1/q}\leq \prod _{i=1}^{n}x_{i}^{w_{i}}\leq \left(\sum _{i=1}^{n}w_{i}x_{i}^{q}\right)^{1/q}.}

The proof follows from [Jensen's inequality](/source/Jensen's_inequality), making use of the fact the [logarithm](/source/Logarithm) is concave: log ⁡ ∏ i = 1 n x i w i = ∑ i = 1 n w i log ⁡ x i ≤ log ⁡ ∑ i = 1 n w i x i . {\displaystyle \log \prod _{i=1}^{n}x_{i}^{w_{i}}=\sum _{i=1}^{n}w_{i}\log x_{i}\leq \log \sum _{i=1}^{n}w_{i}x_{i}.}

By applying the [exponential function](/source/Exponential_function) to both sides and observing that as a strictly increasing function it preserves the sign of the inequality, we get ∏ i = 1 n x i w i ≤ ∑ i = 1 n w i x i . {\displaystyle \prod _{i=1}^{n}x_{i}^{w_{i}}\leq \sum _{i=1}^{n}w_{i}x_{i}.}

Taking q-th powers of the xi yields ∏ i = 1 n x i q ⋅ w i ≤ ∑ i = 1 n w i x i q ∏ i = 1 n x i w i ≤ ( ∑ i = 1 n w i x i q ) 1 / q . {\displaystyle {\begin{aligned}&\prod _{i=1}^{n}x_{i}^{q{\cdot }w_{i}}\leq \sum _{i=1}^{n}w_{i}x_{i}^{q}\\&\prod _{i=1}^{n}x_{i}^{w_{i}}\leq \left(\sum _{i=1}^{n}w_{i}x_{i}^{q}\right)^{1/q}.\end{aligned}}}

Thus, we are done for the inequality with positive q; the case for negatives is identical but for the swapped signs in the last step:

∏ i = 1 n x i − q ⋅ w i ≤ ∑ i = 1 n w i x i − q . {\displaystyle \prod _{i=1}^{n}x_{i}^{-q{\cdot }w_{i}}\leq \sum _{i=1}^{n}w_{i}x_{i}^{-q}.}

Of course, taking each side to the power of a negative number -1/*q* swaps the direction of the inequality.

∏ i = 1 n x i w i ≥ ( ∑ i = 1 n w i x i − q ) − 1 / q . {\displaystyle \prod _{i=1}^{n}x_{i}^{w_{i}}\geq \left(\sum _{i=1}^{n}w_{i}x_{i}^{-q}\right)^{-1/q}.}

### Inequality between any two power means

We are to prove that for any *p* < *q* the following inequality holds: ( ∑ i = 1 n w i x i p ) 1 / p ≤ ( ∑ i = 1 n w i x i q ) 1 / q {\displaystyle \left(\sum _{i=1}^{n}w_{i}x_{i}^{p}\right)^{1/p}\leq \left(\sum _{i=1}^{n}w_{i}x_{i}^{q}\right)^{1/q}} if p is negative, and q is positive, the inequality is equivalent to the one proved above: ( ∑ i = 1 n w i x i p ) 1 / p ≤ ∏ i = 1 n x i w i ≤ ( ∑ i = 1 n w i x i q ) 1 / q {\displaystyle \left(\sum _{i=1}^{n}w_{i}x_{i}^{p}\right)^{1/p}\leq \prod _{i=1}^{n}x_{i}^{w_{i}}\leq \left(\sum _{i=1}^{n}w_{i}x_{i}^{q}\right)^{1/q}}

The proof for positive p and q is as follows: Define the following function: *f* : **R**+ → **R**+ f ( x ) = x q p {\displaystyle f(x)=x^{\frac {q}{p}}} . f is a power function, so it does have a [second derivative](/source/Second_derivative): f ″ ( x ) = ( q p ) ( q p − 1 ) x q p − 2 {\displaystyle f''(x)=\left({\frac {q}{p}}\right)\left({\frac {q}{p}}-1\right)x^{{\frac {q}{p}}-2}} which is strictly positive within the domain of f, since *q* > *p*, so we know f is convex.

Using this, and the Jensen's inequality we get: f ( ∑ i = 1 n w i x i p ) ≤ ∑ i = 1 n w i f ( x i p ) ( ∑ i = 1 n w i x i p ) q / p ≤ ∑ i = 1 n w i x i q {\displaystyle {\begin{aligned}f\left(\sum _{i=1}^{n}w_{i}x_{i}^{p}\right)&\leq \sum _{i=1}^{n}w_{i}f(x_{i}^{p})\\[3pt]\left(\sum _{i=1}^{n}w_{i}x_{i}^{p}\right)^{q/p}&\leq \sum _{i=1}^{n}w_{i}x_{i}^{q}\end{aligned}}} after raising both side to the power of 1/*q* (an increasing function, since 1/*q* is positive) we get the inequality which was to be proven:

( ∑ i = 1 n w i x i p ) 1 / p ≤ ( ∑ i = 1 n w i x i q ) 1 / q {\displaystyle \left(\sum _{i=1}^{n}w_{i}x_{i}^{p}\right)^{1/p}\leq \left(\sum _{i=1}^{n}w_{i}x_{i}^{q}\right)^{1/q}}

Using the previously shown equivalence we can prove the inequality for negative p and q by replacing them with −q and −p, respectively.

## Generalized *f*-mean

Main article: [Generalized f-mean](/source/Generalized_f-mean)

The power mean could be generalized further to the [generalized f-mean](/source/Generalized_f-mean):

M f ( x 1 , … , x n ) = f − 1 ( 1 n ⋅ ∑ i = 1 n f ( x i ) ) {\displaystyle M_{f}(x_{1},\dots ,x_{n})=f^{-1}\left({{\frac {1}{n}}\cdot \sum _{i=1}^{n}{f(x_{i})}}\right)}

This covers the geometric mean without using a limit with *f*(*x*) = log(*x*). The power mean is obtained for *f*(*x*) = *xp*. Properties of these means are studied in de Carvalho (2016).[3]

## Applications

### Signal processing

A power mean serves a non-linear [moving average](/source/Moving_average) which is shifted towards small signal values for small p and emphasizes big signal values for big p. Given an efficient implementation of a [moving arithmetic mean](/source/Lowpass) called smooth one can implement a moving power mean according to the following [Haskell](/source/Haskell_(programming_language)) code.

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

- For big p it can serve as an [envelope detector](/source/Envelope_detector) on a [rectified](/source/Rectifier) signal.

- For small p it can serve as a [baseline detector](https://en.wikipedia.org/w/index.php?title=Baseline_(spectrometry)&action=edit&redlink=1) on a [mass spectrum](/source/Mass_spectrum).

## See also

- [Arithmetic–geometric mean](/source/Arithmetic%E2%80%93geometric_mean)

- [Average](/source/Average)

- [Heronian mean](/source/Heronian_mean)

- [Inequality of arithmetic and geometric means](/source/Inequality_of_arithmetic_and_geometric_means)

- [Lehmer mean](/source/Lehmer_mean) – also a mean related to [powers](/source/Power_(mathematics))

- [Minkowski distance](/source/Minkowski_distance)

- [Quasi-arithmetic mean](/source/Quasi-arithmetic_mean) – another name for the [generalized f-mean](/source/Generalized_f-mean) mentioned above

- [Root mean square](/source/Root_mean_square)

## Notes

1. **[^](#cite_ref-5)** If NM = *a* and PM = *b*. AM = **AM** of *a* and *b*, and radius *r* = AQ = AG. Using [Pythagoras' theorem](/source/Pythagoras'_theorem), QM² = AQ² + AM² ∴ QM = √AQ² + AM² = **QM**. Using Pythagoras' theorem, AM² = AG² + GM² ∴ GM = √AM² − AG² = **GM**. Using [similar triangles](/source/Similar_triangles), ⁠HM/GM⁠ = ⁠GM/AM⁠ ∴ HM = ⁠GM²/AM⁠ = **HM**.

## References

1. ^ [***a***](#cite_ref-sykora_1-0) [***b***](#cite_ref-sykora_1-1) Sýkora, Stanislav (2009). "Mathematical means and averages: basic properties". *Stan's Library*. **III**. Castano Primo, Italy. [doi](/source/Doi_(identifier)):[10.3247/SL3Math09.001](https://doi.org/10.3247%2FSL3Math09.001).

1. ^ [***a***](#cite_ref-Bullen1_2-0) [***b***](#cite_ref-Bullen1_2-1) [***c***](#cite_ref-Bullen1_2-2) P. S. Bullen: *Handbook of Means and Their Inequalities*. Dordrecht, Netherlands: Kluwer, 2003, pp. 175-177

1. ^ [***a***](#cite_ref-dC2016_3-0) [***b***](#cite_ref-dC2016_3-1) de Carvalho, Miguel (2016). ["Mean, what do you Mean?"](https://zenodo.org/record/895400). *[The American Statistician](/source/The_American_Statistician)*. **70** (3): 764‒776. [doi](/source/Doi_(identifier)):[10.1080/00031305.2016.1148632](https://doi.org/10.1080%2F00031305.2016.1148632). [hdl](/source/Hdl_(identifier)):[20.500.11820/fd7a8991-69a4-4fe5-876f-abcd2957a88c](https://hdl.handle.net/20.500.11820%2Ffd7a8991-69a4-4fe5-876f-abcd2957a88c).

1. **[^](#cite_ref-4)** *Handbook of Means and Their Inequalities (Mathematics and Its Applications)*.

## Further reading

- Bullen, P. S. (2003). "Chapter III - The Power Means". *Handbook of Means and Their Inequalities*. Dordrecht, Netherlands: Kluwer. pp. 175–265.

## External links

- [Power mean at MathWorld](https://mathworld.wolfram.com/PowerMean.html)

- [Examples of Generalized Mean](https://people.revoledu.com/kardi/tutorial/BasicMath/Average/Generalized%20mean.html)

- A [proof of the Generalized Mean](https://planetmath.org/ProofOfGeneralMeansInequality) on [PlanetMath](/source/PlanetMath)

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