generalized mean

Definition

If p is a non-zero real number, and

x

1

,

…

,

x

n

{\displaystyle x_{1},\dots ,x_{n}}

are positive real numbers, then the generalized mean or power mean with exponent p of these positive real numbers is

M

p

(

x

1

,

…

,

x

n

)

=

(

1

n

∑

i

=

1

n

x

i

p

)

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.

Properties

Let

x

1

,

…

,

x

n

{\displaystyle x_{1},\dots ,x_{n}}

be a sequence of positive real numbers, then the following properties hold:

min

(

x

1

,

…

,

x

n

)

≤

M

p

(

x

1

,

…

,

x

n

)

≤

max

(

x

Generalized mean inequality

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 x1 = x2 = ... = 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

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:

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}}

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

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:

Generalized f-mean

The power mean could be generalized further to the 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).

Applications

Signal processing

A power mean serves a non-linear 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 called smooth one can implement a moving power mean according to the following Haskell code.

For big p it can serve as an envelope detector on a rectified signal.

For small p it can serve as a baseline detector on a mass spectrum.