Generalized mean

If is a non-zero real number, and x_1, \dots, x_n are positive real numbers, then the generalized mean or power mean with exponent of these positive real numbers is M_p(x_1,\dots,x_n) = \left( \frac{1}{n} \sum_{i=1}^n x_i^p \right)^{{1}/{p}} . (See -norm). For 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, \dots, x_n) = \left(\prod_{i=1}^n x_i\right)^{1/n} . Furthermore, for a sequence of positive weights we define the weighted power mean as 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 , it is equal to the weighted geometric mean: 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 . == Special cases ==

For some values of p, the mean M_p(x_1, \dots, x_n) corresponds to a well known mean. {{Math proof|title=Proof of \lim_{p \to 0} M_p = M_0 (geometric mean)|proof=For the purpose of the proof, we will assume without loss of generality that w_i \in [0,1] and \sum_{i=1}^n w_i = 1. We can rewrite the definition of M_p using the exponential function as M_p(x_1,\dots,x_n) = \exp{\left( \ln{\left[\left(\sum_{i=1}^n w_ix_{i}^p \right)^{1/p}\right]} \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 argument of the exponential function. We assume that p \isin \mathbb{R} but , and that the sum of is equal to 1 (without loss in generality); Differentiating the numerator and denominator with respect to , we have \begin{align} \lim_{p \to 0} \frac{\ln{\left(\sum_{i=1}^n w_ix_{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{align} 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|title= Proof of \lim_{p \to \infty} M_p = M_\infty and \lim_{p \to -\infty} M_p = M_{-\infty} |proof= Assume (possibly after relabeling and combining terms together) that x_1 \geq \dots \geq x_n. Then \begin{align} \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{align} The formula for M_{-\infty} follows from M_{-\infty} (x_1,\dots,x_n) = \frac{1}{M_\infty (1/x_1,\dots,1/x_n)} = x_n. }} ==Properties==

Let x_1, \dots, x_n be a sequence of positive real numbers, then the following properties hold: • \min(x_1, \dots, x_n) \le M_p(x_1, \dots, x_n) \le \max(x_1, \dots, x_n). • M_p(x_1, \dots, x_n) = M_p(P(x_1, \dots, x_n)), where P is a permutation operator. • M_p(b x_1, \dots, b x_n) = b \cdot M_p(x_1, \dots, x_n). • 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]. Generalized mean inequality In general, if , then M_p(x_1, \dots, x_n) \le M_q(x_1, \dots, x_n) and the two means are equal if and only if . The inequality is true for real values of and , as well as positive and negative infinity values. It follows from the fact that, for all real , \frac{\partial}{\partial p}M_p(x_1, \dots, x_n) \geq 0 which can be proved using Jensen's inequality. In particular, for in {{math|{−1, 0, 1}}}, the generalized mean inequality implies the Pythagorean means inequality as well as the 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: \begin{align} w_i \in [0, 1] \\ \sum_{i=1}^nw_i = 1 \end{align} The proof for unweighted power means can be easily obtained by substituting . Equivalence of inequalities between means of opposite signs Suppose an average between power means with exponents and holds: \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: \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): \left(\sum_{i=1}^nw_ix_i^{-p}\right)^{-1/p} = \left(\frac{1}{\sum_{i=1}^nw_i\frac{1}{x_i^p}}\right)^{1/p} \leq \left(\frac{1}{\sum_{i=1}^nw_i\frac{1}{x_i^q}}\right)^{1/q} = \left(\sum_{i=1}^nw_ix_i^{-q}\right)^{-1/q} We get the inequality for means with exponents and , 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 and non-negative weights summing to 1, the following inequality holds: \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, making use of the fact the logarithm is concave: \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 to both sides and observing that as a strictly increasing function it preserves the sign of the inequality, we get \prod_{i=1}^n x_i^{w_i} \leq \sum_{i=1}^n w_i x_i. Taking -th powers of the yields \begin{align} &\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{align} Thus, we are done for the inequality with positive ; the case for negatives is identical but for the swapped signs in the last step: \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 swaps the direction of the inequality. \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 the following inequality holds: \left(\sum_{i=1}^n w_i x_i^p\right)^{1/p} \leq \left(\sum_{i=1}^nw_ix_i^q\right)^{1/q} if is negative, and is positive, the inequality is equivalent to the one proved above: \left(\sum_{i=1}^nw_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 and is as follows: Define the following function: f(x)=x^{\frac{q}{p}}. is a power function, so it does have a second derivative: 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 , since , so we know is convex. Using this, and the Jensen's inequality we get: \begin{align} f \left( \sum_{i=1}^nw_ix_i^p \right) &\leq \sum_{i=1}^nw_if(x_i^p) \\[3pt] \left(\sum_{i=1}^n w_i x_i^p\right)^{q/p} &\leq \sum_{i=1}^nw_ix_i^q \end{align} after raising both side to the power of (an increasing function, since is positive) we get the inequality which was to be proven: \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 and by replacing them with and , respectively. == Generalized f-mean ==

The power mean could be generalized further to the generalized -mean: 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 . The power mean is obtained for . 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 and emphasizes big signal values for big . Given an efficient implementation of a moving arithmetic mean called smooth one can implement a moving power mean according to the following Haskell code. powerSmooth :: Floating a => ([a] -> [a]) -> a -> [a] -> [a] powerSmooth smooth p = map (** recip p) . smooth . map (**p) • For big it can serve as an envelope detector on a rectified signal. • For small it can serve as a baseline detector on a mass spectrum. ==See also==