Generalized Mean - Definition

Definition

If is a non-zero real number, we can define the generalized mean or power mean with exponent of the positive real numbers as:

M_p(x_1,\dots,x_n) = \left( \frac{1}{n} \sum_{i=1}^n x_i^p \right)^{1/p}

While for we assume that it's equal to the geometric mean (which is, in fact, the limit of means with exponents approaching zero, as proved below for the general case):

M_0(x_1, \dots, x_n) = \sqrt{\prod_{i=1}^n x_i}

Furthermore, for a sequence of positive weights with sum we define the weighted power mean as:

M_p(x_1,\dots,x_n) = \left(\sum_{i=1}^n w_i x_i^p \right)^{1/p}
M_0(x_1,\dots,x_n) = \prod_{i=1}^n x_i^{w_i}

The unweighted means correspond to setting all . For exponents equal to positive or negative infinity the means are maximum and minimum, respectively, regardless of weights (and they are actually the limit points for exponents approaching the respective extremes, as proved below):

M_\infty (x_1,\dots,x_n)=\max(x_1,\dots,x_n)
M_{-\infty}(x_1,\dots,x_n)=\min(x_1,\dots,x_n)
Proof that (geometric mean)

We can rewrite the definition of using the exponential function

M_p(x_1,\dots,x_n) = \exp{\left( \ln{\left(\sum_{i=1}^n w_ix_{i}^p \right)^{1/p}} \right) } = \exp{\left( \frac{\ln{\left(\sum_{i=1}^n w_ix_{i}^p \right)}}{p} \right) }

In the limit, we can apply L'Hôpital's rule to the exponential component,

\lim_{p \to 0} \frac{\ln{\left(\sum_{i=1}^n w_ix_{i}^p \right)}}{p} = \lim_{p \to 0} \frac{\sum_{i=1}^n w_i x_i^p \ln{x_i}}{\sum_{i=1}^n w_i x_i^p} = \sum_{i=1}^n w_i \ln{x_i} = \ln{\left(\prod_{i=1}^n x_i^{w_i} \right)}

since . By the continuity of the exponential function, we can substitute back into the above relation to obtain

\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.
Proof that

Assume (possibly after relabeling and combining terms together) that . Then

\lim_{p \to \infty} \sum_{i=1}^n w_i x_i^p = \lim_{p \to \infty} x_1^p \sum_{i=1}^n w_i \left( \frac{x_i}{x_1} \right)^p = \lim_{p \to \infty} w_1 x_1^p

because for . (Note that although for goes to infinity, it does so much slower than so that in the limit it can be ignored.) Thus

\lim_{p \to \infty} M_p(x_1,\dots,x_n) = \lim_{p \to \infty} \left(w_1 x_1^p \right)^{1/p} = x_1 = \max(x_1,\dots,x_n) = M_\infty (x_1,\dots,x_n)

since . The formula for follows from .

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