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Generalized mean

The generalized mean, also known as the power mean or Hölder mean, is a family of means that extends common statistical measures of central tendency, such as the arithmetic, geometric, and harmonic means, to a parameterized form for positive real numbers a_1, a_2, \dots, a_n and a real parameter p \neq 0, defined by the formula

M_p(a_1, \dots, a_n) = \left( \frac{1}{n} \sum_{k=1}^n a_k^p \right)^{1/p}.

^[1] For p = 0, the generalized mean is defined as the limit

M_0(a_1, \dots, a_n) = \lim_{p \to 0} M_p(a_1, \dots, a_n) = \left( \prod_{k=1}^n a_k \right)^{1/n},

corresponding to the geometric mean.^[1] This construction unifies various means under a single framework, where the choice of p determines the type of mean, with p > 1 emphasizing larger values and p < 1 emphasizing smaller ones.^[2] Key special cases of the generalized mean include the minimum (p \to -\infty), harmonic mean (p = -1), geometric mean (p = 0), arithmetic mean (p = 1), root mean square or quadratic mean (p = 2), and maximum (p \to \infty).^[1] These are summarized in the following table: ^[1] The generalized mean also admits weighted versions, where equal weights $1/n are replaced by positive weights p_i summing to 1, further broadening its applicability in statistics and analysis.^[2] A fundamental property is the power mean inequality, which states that for p < q and positive a_k not all equal, M_p(a_1, \dots, a_n) \leq M_q(a_1, \dots, a_n), with equality if and only if all a_k are identical; this generalizes classical inequalities like the arithmetic mean-geometric mean inequality.^[1] The concept traces back to early 20th-century work in inequalities and has applications in optimization, signal processing, and information theory.^[1]

Definition and Formulation

Power Mean Definition

The power mean of order p, denoted M_p(x_1, \dots, x_n), is a family of means defined for positive real numbers x_1, \dots, x_n > 0 and real exponent p \neq 0 by the formula

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

^[1] This expression aggregates the values by raising them to the power p, averaging, and then taking the p-th root, providing a parameterized way to measure central tendency that varies with p. For p = 0, the power mean is defined as the limit

M_0(x_1, \dots, x_n) = \lim_{p \to 0} M_p(x_1, \dots, x_n) = \exp\left( \frac{1}{n} \sum_{i=1}^n \ln x_i \right),

which yields the geometric mean.^[1] The requirement that all x_i > 0 ensures well-definedness, avoiding issues with negative bases in non-integer powers or complex roots for even denominators.^[1] For p > 0, the power mean admits an interpretation in terms of the \ell_p-norm of the vector (x_1, \dots, x_n), specifically M_p(x_1, \dots, x_n) = \|(x_1, \dots, x_n)\|_p / n^{1/p}, where \|(x_1, \dots, x_n)\|_p = \left( \sum_{i=1}^n x_i^p \right)^{1/p}.^[1] This connection highlights the power mean as a normalized version of the \ell_p-norm, scaling it to account for the number of elements and thus behaving like an average. The concept of power means emerged in the context of inequalities in the 1920s and was systematically explored in the seminal work Inequalities by G. H. Hardy, J. E. Littlewood, and G. Pólya, published in 1934, where they analyzed sequences of such means and their limiting behaviors.

Weighted Variants

The weighted power mean extends the concept of the power mean to account for varying importance of individual data points by incorporating positive weights w_i > 0.^[3]^[4] For p \neq 0, it is defined by the formula

M_p(\mathbf{w}; 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},

where x_1, \dots, x_n > 0 are the data points.^[3] This form accommodates weights summing to any positive total, as the ratio ensures scale invariance.^[4] Due to the homogeneity property of the power mean—where scaling all x_i by a constant c > 0 scales the mean by c—the weights can be normalized to sum to 1 without altering the result.^[3] When all w_i are equal, the weighted power mean reduces to the unweighted case from the standard power mean definition.^[4] For the limiting case p = 0, the weighted power mean is the weighted geometric mean, obtained as

M_0(\mathbf{w}; x_1, \dots, x_n) = \exp\left( \frac{\sum_{i=1}^n w_i \log x_i}{\sum_{i=1}^n w_i} \right).

^[3] In statistical contexts, the weights w_i typically represent frequencies (indicating the number of occurrences of each x_i) or probabilities (normalized to sum to 1, reflecting relative likelihoods).^[5]

Special Cases and Examples

Arithmetic, Geometric, and Harmonic Means

The arithmetic mean, corresponding to the power mean with exponent p = 1, is defined for a set of positive real numbers x_1, x_2, \dots, x_n as

M_1 = \frac{1}{n} \sum_{i=1}^n x_i.

It represents the ordinary average of the values and is widely used in statistics as a measure of central tendency.^[6]^[1] The geometric mean arises as the special case with p = 0, obtained as the limit

M_0 = \lim_{p \to 0} M_p = \exp\left( \frac{1}{n} \sum_{i=1}^n \ln x_i \right) = \left( \prod_{i=1}^n x_i \right)^{1/n},

provided all x_i > 0. This formulation interprets it as the exponential of the average of the logarithms, making it suitable for averaging ratios or growth rates.^[7]^[1] The harmonic mean corresponds to p = -1 and is given by

M_{-1} = \left( \frac{1}{n} \sum_{i=1}^n \frac{1}{x_i} \right)^{-1} = \frac{n}{\sum_{i=1}^n \frac{1}{x_i}},

again for positive x_i. It equals the reciprocal of the average of the reciprocals and is particularly appropriate for averaging rates or ratios, such as speeds over equal distances.^[8]^[1] For positive real numbers, these means satisfy the inequality M_1 \geq M_0 \geq M_{-1}, with equality if and only if all x_i are equal.^[9]^[1] As an illustration, consider the values x = \{1, 2, [3](/page/3)\}. The arithmetic mean is ([1](/page/1) + 2 + [3](/page/3))/3 = 2, the geometric mean is ([1](/page/1) \cdot 2 \cdot [3](/page/3))^{1/3} = 6^{1/3} \approx 1.817, and the harmonic mean is $3 / ([1](/page/1)/[1](/page/1) + 1/2 + 1/[3](/page/3)) = 3 / (11/6) = 18/11 \approx 1.636, confirming $2 > 1.817 > 1.636.^[6]^[7]^[8]

Quadratic and Higher-Order Means

The quadratic mean, corresponding to the power mean with exponent p = 2, is defined as

M_2(\mathbf{x}) = \sqrt{\frac{1}{n} \sum_{i=1}^n x_i^2},

where \mathbf{x} = (x_1, \dots, x_n) is a vector of non-negative real numbers. Also known as the root-mean-square (RMS), this measure quantifies the magnitude of a set of values by emphasizing their squared contributions before averaging and rooting.^[1] In statistics, the population standard deviation \sigma relates closely to the quadratic mean, specifically as the quadratic mean applied to the deviations from the arithmetic mean: \sigma = \sqrt{\frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^2}, where \bar{x} is the arithmetic mean; this connection highlights its role in measuring dispersion around the central tendency.^[10] For higher exponents such as p = 3, the cubic mean is given by

M_3(\mathbf{x}) = \left( \frac{1}{n} \sum_{i=1}^n x_i^3 \right)^{1/3}.

This extends the power mean framework, with the general form for p \geq 2 being M_p(\mathbf{x}) = \left( \frac{1}{n} \sum_{i=1}^n x_i^p \right)^{1/p}. As p increases beyond 1, these means place progressively greater weight on larger values in the dataset, resulting in heightened sensitivity to outliers or extreme observations.^[1] For instance, while the quadratic mean balances magnitudes quadratically, the cubic and higher-order means amplify disparities even further, making them suitable for applications where dominant values drive the overall assessment, such as in signal processing or reliability analysis. In the context of vector spaces, the quadratic mean connects directly to norm theory: for a vector \mathbf{x} \in \mathbb{R}^n, M_2(\mathbf{x}) equals the Euclidean norm \|\mathbf{x}\|_2 divided by \sqrt{n}, underscoring its utility in geometry and physics for computing effective lengths or energies.^[1] Due to the monotonicity of power means with respect to the exponent, the quadratic mean exceeds the arithmetic mean for any dataset exhibiting positive variance.^[1]

Limiting Cases for Extreme Exponents

As the exponent p in the power mean M_p(\mathbf{x}) = \left( \frac{1}{n} \sum_{i=1}^n x_i^p \right)^{1/p} (for positive x_i) approaches positive infinity, the mean converges to the maximum value among the x_i. To demonstrate this, let M = \max_i x_i. The upper bound follows immediately: since each x_i \leq M, \sum x_i^p \leq n M^p, so M_p \leq M. For the lower bound, the sum includes at least one term equal to M^p, so \sum x_i^p \geq M^p, yielding M_p \geq M \cdot n^{-1/p}. As p \to \infty, n^{-1/p} \to 1, and by the squeeze theorem, M_p \to M. This result extends to weighted power means via similar normalization arguments. Symmetrically, as p \to -\infty, M_p converges to the minimum value m = \min_i x_i. Substituting q = -p (so q \to \infty), the expression becomes M_{-q} = \left( \frac{1}{n} \sum x_i^{-q} \right)^{-1/q}. The inner term \left( \frac{1}{n} \sum x_i^{-q} \right)^{1/q} follows the same limiting behavior as above but applied to the reciprocals $1/x_i, approaching \max_i (1/x_i) = 1/m. Thus, M_{-q} \to m. Again, this holds for weighted variants by adjusting the weights in the dominance argument. An intermediate limit occurs as p \to 0, where M_p approaches the geometric mean G(\mathbf{x}) = \left( \prod_{i=1}^n x_i \right)^{1/n}. This can be confirmed by considering the logarithm: p \log M_p = \log \left( \frac{1}{n} \sum x_i^p \right), and applying L'Hôpital's rule to the indeterminate form as p \to 0 yields \frac{1}{n} \sum \log x_i = \log G(\mathbf{x}), so M_p \to G(\mathbf{x}). These extreme limiting cases are particularly useful in optimization contexts where the objective function or performance metric is dominated by the largest or smallest elements, such as in minimax problems or robust aggregation methods that emphasize outliers.

Mathematical Properties

Homogeneity and Symmetry

The power mean of order p, denoted M_p(\mathbf{x}) for a vector of positive real numbers \mathbf{x} = (x_1, \dots, x_n), exhibits homogeneity of degree 1. Specifically, for any \lambda > 0, M_p(\lambda \mathbf{x}) = \lambda M_p(\mathbf{x}).^[11] This property arises directly from the formulation, where scaling the inputs by \lambda scales the p-th powers by \lambda^p and the subsequent root by \lambda, preserving the overall degree.^[11] In the weighted case, with weights w_i > 0 summing to 1, the weighted power mean M_p(\mathbf{x}, \mathbf{w}) retains this homogeneity: M_p(\lambda \mathbf{x}, \mathbf{w}) = \lambda M_p(\mathbf{x}, \mathbf{w}).^[11] The unweighted power mean is symmetric with respect to its arguments, meaning M_p(\mathbf{x}) remains unchanged under any permutation of the x_i.^[11] This invariance follows from the symmetric summation in the definition. For the weighted variant, symmetry holds under simultaneous permutations of the x_i and corresponding w_i, ensuring the mean depends only on the paired values rather than their order.^[11] For positive integer orders p, the power mean relates directly to integer power sums, as [M_p(\mathbf{x})]^p = \frac{1}{n} \sum_{i=1}^n x_i^p, linking it to the arithmetic mean of the p-th powers. The power mean M_p(\mathbf{x}) is continuous as a function of the exponent p \in \mathbb{R} for fixed positive \mathbf{x}, with the function extending continuously to the boundaries via limits: as p \to 0, M_p \to geometric mean; as p \to \infty, M_p \to \max(\mathbf{x}); and as p \to -\infty, M_p \to \min(\mathbf{x}).

Monotonicity in Exponent

One defining property of the power mean, or generalized mean of order p, is its monotonicity with respect to the exponent p. For a fixed set of positive real numbers x_1, x_2, \dots, x_n > 0 that are not all equal, if p < q, then M_p(x_1, \dots, x_n) \leq M_q(x_1, \dots, x_n), where M_r = \left( \frac{1}{n} \sum_{i=1}^n x_i^r \right)^{1/r} for r \neq 0 (and the geometric mean for r = 0), with strict inequality holding unless all x_i are identical.^[12] This result, a cornerstone of mean inequalities, originates from classical analyses of symmetric convex functions and has been extensively documented in foundational texts on inequalities. The underlying intuition for this ordering stems from the differing influences of the exponent on individual terms: lower values of p relatively amplify the contribution of smaller x_i in the aggregated average, pulling the mean downward, whereas higher p disproportionately boosts the larger x_i, elevating the mean. For instance, as p approaches -\infty, M_p converges to the minimum x_i, emphasizing the smallest value, while as p approaches \infty, it approaches the maximum, highlighting the largest.^[12] This monotonicity extends naturally to the weighted power mean, defined as M_p = \left( \sum_{i=1}^n w_i x_i^p \right)^{1/p} where w_i > 0 are weights summing to 1. Under the same conditions of positive x_i not all equal and p < q, the inequality M_p \leq M_q holds strictly unless all x_i coincide.^[13] The requirement for positivity of the x_i is essential, as the power mean is generally undefined for non-integer p when negative or zero values are present, and the monotonicity may fail in such cases—for example, introducing a negative x_i can reverse the ordering for certain fractional exponents due to complex values or altered convexity.

Generalized Mean Inequality

The generalized mean inequality, often referred to as the power mean inequality, asserts that the power means are monotonically increasing with respect to the exponent parameter. Specifically, for positive real numbers x_1, x_2, \dots, x_n > 0 and real exponents satisfying -\infty \leq p \leq q \leq \infty, the inequality M_p(x_1, \dots, x_n) \leq M_q(x_1, \dots, x_n) holds, where M_r denotes the r-th power mean. Equality occurs if and only if all the x_i are equal.^[1]^[14] This result extends naturally to the weighted case. For nonnegative weights w_1, w_2, \dots, w_n \geq 0 with \sum_{i=1}^n w_i = 1 and the same conditions on the positive x_i and exponents p \leq q, the weighted power means satisfy M_p^w(x_1, \dots, x_n) \leq M_q^w(x_1, \dots, x_n), where M_r^w = \left( \sum_{i=1}^n w_i x_i^r \right)^{1/r} for finite r, with appropriate limits for r = 0, \pm \infty. Again, equality holds if and only if all x_i are equal.^[14] The inequality was formalized and proved in the influential 1934 monograph Inequalities by G. H. Hardy, J. E. Littlewood, and G. Pólya, which built upon earlier specific cases such as the arithmetic-geometric mean inequality to provide a unified framework for means and their ordering.^[15] This work filled a historical gap by systematically addressing inequalities among various means, influencing subsequent developments in analysis and inequalities.^[14] The power mean inequality serves as a foundational result that extends to broader classes, such as quasi-arithmetic means defined via convex functions, where analogous monotonicity properties hold under suitable conditions.

Proofs and Theoretical Foundations

Derivation of the Inequality

The generalized mean inequality asserts that for real numbers a_1, a_2, \dots, a_n > 0 and weights w_1, w_2, \dots, w_n > 0 with \sum_{i=1}^n w_i = 1, if p < q, then M_p(\mathbf{a}; \mathbf{w}) \leq M_q(\mathbf{a}; \mathbf{w}), where M_r(\mathbf{a}; \mathbf{w}) = \left( \sum_{i=1}^n w_i a_i^r \right)^{1/r} for r \neq 0, with equality if and only if a_1 = a_2 = \dots = a_n.^[16] For the case $0 < p < q, the proof relies on Jensen's inequality applied to the convex function \phi(t) = t^{q/p}, which is convex on [0, \infty) since q/p > 1. Without loss of generality, normalize the variables by setting x_i = a_i / M_p(\mathbf{a}; \mathbf{w}) for each i, so that \sum_{i=1}^n w_i x_i^p = 1 and x_i \geq 0. Then, M_q(\mathbf{a}; \mathbf{w}) = M_p(\mathbf{a}; \mathbf{w}) \cdot \left( \sum_{i=1}^n w_i x_i^q \right)^{1/q}. Substituting y_i = x_i^p, it follows that \sum_{i=1}^n w_i y_i = 1 and y_i \geq 0, with x_i^q = y_i^{q/p}. By Jensen's inequality, \sum_{i=1}^n w_i \phi(y_i) \geq \phi\left( \sum_{i=1}^n w_i y_i \right), so \sum_{i=1}^n w_i y_i^{q/p} \geq 1^{q/p} = 1. Hence \sum_{i=1}^n w_i x_i^q \geq 1, implying M_q(\mathbf{a}; \mathbf{w}) \geq M_p(\mathbf{a}; \mathbf{w}). Equality holds if and only if all y_i are equal, which occurs precisely when all a_i are equal, due to the strict convexity of \phi.^[16]^[15] The weighted case follows directly from the weighted form of Jensen's inequality, which replaces the uniform average with the weighted average \sum w_i f(y_i) \geq f\left( \sum w_i y_i \right) for convex f, under the same normalization and convexity assumptions as above.^[15] For cases involving the geometric mean (where p = 0), the inequality M_0(\mathbf{a}; \mathbf{w}) \leq M_q(\mathbf{a}; \mathbf{w}) for q > 0 follows from the log-convexity of the function t \mapsto t^q, or equivalently, from the concavity of the logarithm applied to the power means. Specifically, since \ln M_q(\mathbf{a}; \mathbf{w}) = \frac{1}{q} \ln \left( \sum_{i=1}^n w_i a_i^q \right) and the function u \mapsto \ln u is concave, Jensen's inequality yields \sum_{i=1}^n w_i \ln (a_i^q) \leq \ln \left( \sum_{i=1}^n w_i a_i^q \right), so q \sum_{i=1}^n w_i \ln a_i \leq q \ln M_q(\mathbf{a}; \mathbf{w}), hence \ln M_0(\mathbf{a}; \mathbf{w}) \leq \ln M_q(\mathbf{a}; \mathbf{w}), implying the desired inequality. Equality again holds if and only if all a_i are equal. The case p < 0 < q combines the above with the inequality for negative exponents via reciprocity relations.^[15] An alternative derivation for specific cases, such as the arithmetic-geometric mean inequality, employs Hölder's inequality: for p = q/(q-p) and conjugate p' = q/p, it bounds \sum w_i a_i \cdot 1 \leq \left( \sum w_i a_i^q \right)^{1/q} \left( \sum w_i \right)^{1 - 1/q}, simplifying to M_1 \leq M_q under normalization. However, the convexity-based approach via Jensen's inequality provides the primary and most general framework.^[17]

Limit Behaviors and Equivalences

As the exponent p approaches infinity, the generalized mean M_p(\mathbf{x}) of a finite set of positive real numbers \mathbf{x} = (x_1, \dots, x_n) converges to the maximum value among them. To see this, let m = \max\{x_1, \dots, x_n\} and assume without loss of generality that x_1 = m \geq x_i for all i > 1. Then,

M_p(\mathbf{x}) = m \left( \frac{1}{n} \sum_{i=1}^n \left( \frac{x_i}{m} \right)^p \right)^{1/p}.

As p \to \infty, \left( \frac{x_i}{m} \right)^p \to 0 for each i with x_i < m, while it remains 1 for those i where x_i = m. Suppose there are k \geq 1 such maxima; the sum approaches \frac{k}{n}, and \left( \frac{k}{n} \right)^{1/p} \to 1, yielding M_p(\mathbf{x}) \to m.^[18] Similarly, as p \to -\infty, M_p(\mathbf{x}) converges to the minimum value \min\{x_1, \dots, x_n\}. This follows by symmetry: letting q = -p > 0, as q \to \infty, M_{-q}(\mathbf{x}) = \left( \frac{1}{n} \sum_{i=1}^n x_i^{-q} \right)^{-1/q} is the reciprocal of the q-th generalized mean of the reciprocals \{1/x_i\}, which approaches the reciprocal of the maximum of \{1/x_i\}, or equivalently the minimum of \{x_i\}.^[18] The generalized means also exhibit equivalences relating exponents of opposite signs. Specifically, for p > 0, the -p-th mean satisfies M_{-p}(\mathbf{x}) = \left( M_p(1/\mathbf{x}) \right)^{-1}, where $1/\mathbf{x} = (1/x_1, \dots, 1/x_n). This relation arises directly from the definition:

M_{-p}(\mathbf{x}) = \left( \frac{1}{n} \sum_{i=1}^n x_i^{-p} \right)^{-1/p} = \left( \frac{1}{n} \sum_{i=1}^n (1/x_i)^{p} \right)^{-1/p} = \left( M_p(1/\mathbf{x}) \right)^{-1}.

A classic example is the harmonic mean (p = -1), which equals the reciprocal of the arithmetic mean (p = 1) of the reciprocals.^[18] The case p = 0 is defined as the limit \lim_{p \to 0} M_p(\mathbf{x}), which equals the geometric mean G(\mathbf{x}) = ( \prod_{i=1}^n x_i )^{1/n}. To derive this, consider the natural logarithm:

\ln M_p(\mathbf{x}) = \frac{1}{p} \ln \left( \frac{1}{n} \sum_{i=1}^n x_i^p \right).

As p \to 0, this is an indeterminate form \frac{0}{0}. Applying L'Hôpital's rule, differentiate the numerator and denominator with respect to p:

\frac{d}{dp} \left[ \ln \left( \frac{1}{n} \sum_{i=1}^n x_i^p \right) \right] = \frac{ \sum_{i=1}^n x_i^p \ln x_i }{ \sum_{i=1}^n x_i^p }, \quad \frac{d}{dp} = 1.

Evaluating at p = 0 gives \frac{1}{n} \sum_{i=1}^n \ln x_i = \ln G(\mathbf{x}), so \ln M_p(\mathbf{x}) \to \ln G(\mathbf{x}) and thus M_p(\mathbf{x}) \to G(\mathbf{x}).^[18]

Jensen's Inequality Connection

The power mean of order p, denoted M_p(\mathbf{x}), for positive real numbers x_1, \dots, x_n and equal weights, is defined as M_p(\mathbf{x}) = \left( \frac{1}{n} \sum_{i=1}^n x_i^p \right)^{1/p} for p \neq 0. This expression aligns with the framework of quasi-arithmetic means, where M_f(\mathbf{x}) = f^{-1} \left( \frac{1}{n} \sum_{i=1}^n f(x_i) \right) and the generator function is f(t) = \frac{t^p}{p} for p \neq 0.^[19] The choice of this generator ensures the representation matches the standard power mean, as the constant factor $1/p scales linearly without altering the inverse structure.^[19] For p > 1, the function f(t) = \frac{t^p}{p} is convex on (0, \infty), since its second derivative is (p-1) t^{p-2} > 0.^[19] Jensen's inequality then applies directly: \frac{1}{n} \sum_{i=1}^n f(x_i) \geq f\left( \frac{1}{n} \sum_{i=1}^n x_i \right), which substitutes to f(M_p(\mathbf{x})) \geq f(M_1(\mathbf{x})). Since f is strictly increasing, it follows that M_p(\mathbf{x}) \geq M_1(\mathbf{x}), with equality if and only if all x_i are equal.^[19] For |p| > 1 with appropriate adjustments (e.g., considering concavity for $0 < p < 1 or behavior for negative p), similar convexity arguments extend the inequality framework to other orders.^[19] This connection positions all power means as special cases of quasi-arithmetic means, where the convexity of the generator f underpins monotonicity and inequality properties via Jensen's inequality.^[20]

Generalizations and Extensions

Quasi-Arithmetic Means

Quasi-arithmetic means generalize the concept of power means by allowing for a broader class of transformations through continuous strictly monotonic functions. For positive real numbers x_1, x_2, \dots, x_n and a continuous strictly increasing function f: \mathbb{R}^+ \to \mathbb{R}, the quasi-arithmetic mean M_f is defined as

M_f(x_1, \dots, x_n) = f^{-1}\left( \frac{1}{n} \sum_{i=1}^n f(x_i) \right),

where f^{-1} denotes the inverse function of f. This construction, also known as the Kolmogorov mean, was independently characterized by Kolmogorov and Nagumo in 1930 as a fundamental form of averaging that preserves order under monotonic transformations.^[21] Power means emerge as special cases of quasi-arithmetic means when the generating function f takes specific power forms. For p \neq 0, setting f(t) = t^p yields the power mean of order p, M_p(x_1, \dots, x_n) = \left( \frac{1}{n} \sum_{i=1}^n x_i^p \right)^{1/p}, while for p = 0, the limit case corresponds to f(t) = \log t, recovering the geometric mean \left( \prod_{i=1}^n x_i \right)^{1/n}. These connections highlight how quasi-arithmetic means encompass the family of power means while extending their applicability to non-power transformations.^[21] A key property of quasi-arithmetic means is their ability to preserve certain inequalities when the generating function f satisfies convexity conditions, drawing on Jensen's inequality for convex functions. Specifically, if f is convex, then M_f \geq M_g for another generator g under appropriate majorization or ordering, ensuring monotonicity in the choice of f. For instance, the arithmetic mean-geometric mean inequality (AM-GM) arises by taking f(t) = \log t, which is concave, leading to M_{\log} \leq M_{\mathrm{id}} where M_{\mathrm{id}} is the arithmetic mean, as the concavity of \log implies the product of exponentials averages below the exponential of the average.^[21] An illustrative example is the exponential mean, generated by f(t) = e^t, which produces

M_f(x_1, \dots, x_n) = \log \left( \frac{1}{n} \sum_{i=1}^n e^{x_i} \right).

This mean is particularly useful in contexts involving exponential growth or logarithmic scales, as it weights larger values more heavily due to the convexity of the exponential function.^[21]

Generalized f-Means

The generalized f-mean, also known as the quasi-arithmetic mean or Kolmogorov-Nagumo mean, is defined for positive real numbers x_1, \dots, x_n > 0 and a continuous strictly increasing function f as

M_f(x_1, \dots, x_n) = f^{-1}\left( \frac{1}{n} \sum_{i=1}^n f(x_i) \right).

This formulation accommodates a wide range of transformations, enabling applications where data exhibit behaviors captured by monotonic functions.^[22] A key property is its relation to power means: when f(t) = t^p for p \neq 0 and f is strictly increasing (e.g., p > 0), the generalized f-mean coincides with the power mean M_p = \left( \frac{1}{n} \sum_{i=1}^n x_i^p \right)^{1/p}, while for p = 0, it aligns with the geometric mean via f(t) = \log t. This subset connection highlights how the generalized f-mean encompasses familiar cases while offering flexibility for custom f.^[22] Extensions incorporate non-additive measures for advanced scenarios, such as replacing the arithmetic average with a generalized Choquet integral relative to fuzzy measures (capacities \mu and \nu). The resulting quasi-arithmetic type mean is

M_{u,\mu,\nu}(X) = u^{-1} \left( (C)_{\mu,\nu}(u \circ X) \right),

where u is strictly increasing and continuous, and (C)_{\mu,\nu} is the generalized Choquet integral, allowing modeling of interactions and dependencies among data points; this form is generally asymmetric.^[23] In robust statistics, generalized f-means have seen increased application since the 2010s, particularly for handling asymmetric data and outliers through tailored f that limit the influence of extreme values, as in derivations of generalized f-statistics via maximum likelihood for preprocessing and detection tasks.^[24]

Non-Power Extensions

One prominent non-power extension involves ordered power means, which incorporate order statistics to generalize the standard power mean construction. These means apply power transformations to sorted data points, allowing for sensitivity to the distributional order and tail behavior. A key example is the Gini mean of order r and s, defined for positive real numbers x and y as

G_{r,s}(x,y) = \begin{cases} \left( \frac{x^r + y^r}{x^s + y^s} \right)^{\frac{1}{r-s}} & \text{if } r \neq s, \\ \exp\left( \frac{x^r \ln x + y^r \ln y}{x^r + y^r} \right) & \text{if } r = s, \end{cases}

which extends the power mean family by introducing a second parameter s, enabling finer control over weighting based on relative magnitudes akin to order rankings.^[25] This formulation preserves homogeneity and monotonicity properties while allowing the mean to interpolate between common means like arithmetic (r=1, s=0) and harmonic (r=-1, s=0), and it has been characterized as coinciding with power means in specific parameter intersections.^[25] The logarithmic mean provides another extension outside the power framework, positioned strictly between the arithmetic and geometric means for positive reals a > b > 0. Defined as

L(a,b) = \frac{a - b}{\ln a - \ln b},

it arises naturally in contexts like heat transfer and integral approximations, offering a tighter bound than power means for certain inequalities. Generalizations to higher orders, such as the generalized logarithmic mean of order p,

L_p(a,b) = \frac{p(a^p - b^p)}{\sum_{k=0}^{p-1} (a^{p-1-k} b^k - a^k b^{p-1-k})} \quad (p \geq 1),

extend this by incorporating polynomial-like weighting, maintaining the mean's internality and homogeneity while enhancing applicability in monotonicity studies and inequality refinements.^[26] In recent statistical literature, penalized means have emerged as robust extensions to handle outliers, modifying mean objectives with penalty terms to downweight anomalous observations. For instance, in linear models, penalized weighted least squares introduces weights and penalties on mean shifts for robust estimation. These methods are suitable for noisy data in high-dimensional settings.^[24] Matrix generalizations extend power means to positive definite matrices, particularly covariance matrices, where the trace provides a scalar proxy for total variability. The matrix power mean of order p for positive definite matrices A and B interpolates between the harmonic and arithmetic means via operator means on the Riemannian manifold of positive definite matrices, converging to the geometric mean as p \to 0. For covariance matrices \Sigma_1, \Sigma_2, trace-based variants compute the power mean on eigenvalues before reconstructing, yielding a robust aggregator of multivariate dispersion; for example, the trace of the p-th power mean scales the total variance analogously to scalar cases.^[27] This framework preserves monotonicity in p and finds use in quantum information and multivariate analysis.^[28]

Applications and Uses

Signal and Image Processing

In signal and image processing, generalized means, also known as power means, serve as non-linear filters that aggregate signal values over a window to suppress noise while preserving key features. For instance, the arithmetic mean (p=1) performs standard averaging for smoothing, the quadratic mean (p=2) computes root-mean-square energy to quantify signal power, and the limits as p → ±∞ yield the maximum and minimum values, respectively, which can highlight extreme signal features such as peaks or troughs. These p-norm generalizations of the least mean squares algorithm extend linear adaptive filtering to handle non-Gaussian noise effectively, providing tighter error bounds in scenarios with impulsive interference.^[29] Generalized power means extend traditional moving average filters in digital signal processing, replacing arithmetic averaging with higher- or lower-order means to improve robustness against varying noise levels. By adjusting the power parameter p, these filters can prioritize different signal characteristics, such as reducing sensitivity to extreme values in noisy environments through values of p greater than 1, which has been applied in adaptive filtering for echo cancellation and system identification.^[30]^[31] In image processing, the root-mean-square (RMS) contrast, derived from the quadratic mean (p=2), measures local intensity variations to assess image quality and visibility, independent of spatial frequency content, making it suitable for evaluating complex natural scenes. The harmonic mean (p=-1) is employed in edge detection algorithms, where anti-harmonic averaging highlights boundaries by suppressing uniform regions while preserving sharp transitions, outperforming arithmetic means in noisy images. For example, in audio signal processing, the harmonic mean (p=-1) averages rates such as sound speeds in acoustic profiles, providing a reliable estimate for propagation modeling in underwater or room acoustics.^[32]^[33]^[34]

Statistics and Probability

In statistics, generalized means, also known as power means, are widely used as robust estimators of location and central tendency, particularly in scenarios involving transformed data or specific distributional assumptions. They arise naturally as least squares estimators when minimizing a distance function based on a monotone transformation h(x), where the generalized mean is given by M_h = h^{-1}\left(\frac{1}{n} \sum_{i=1}^n h(x_i)\right). For instance, the arithmetic mean corresponds to h(x) = x under squared error loss, while the geometric mean uses h(x) = \log x for multiplicative models. Similarly, they serve as maximum likelihood estimators in exponential family distributions: the arithmetic mean for the normal distribution, the geometric mean for the lognormal, and the harmonic mean for the inverse gamma. This framework highlights their utility in providing intuitive, computationally simple summaries of data while adapting to underlying model structures. In probability theory, the generalized mean extends to random variables through its connection to L_p spaces, where the p-th power mean of a random variable X is defined as \|X\|_p = \left( \mathbb{E}[|X|^p] \right)^{1/p} for $1 \leq p < \infty, representing the L_p-norm. This formulation generalizes classical moments—the first moment aligns with the arithmetic mean (p=1), and higher p emphasize tail behavior—enabling analysis of integrability and concentration properties in stochastic processes. For heavy-tailed distributions like the Cauchy, negative power means (-1 \leq p < 0) yield unbiased, strongly consistent, and \sqrt{n}-consistent estimators for location and scale parameters, with asymptotic variances that decrease as p approaches 0 from below, offering robustness against outliers. Such estimators facilitate confidence regions and parameter recovery in mixture models, such as Cauchy mixtures, using closed-form expressions derived from fractional moments. Generalized means also play a key role in combining dependent statistical tests via generalized mean p-values (GMPs), defined as p_r = \left( \frac{1}{n} \sum_{i=1}^n p_i^r \right)^{1/r} for r \neq 0, which unify methods like the Bonferroni correction (r \to -\infty), harmonic mean (r = -1), and geometric mean (r \to 0). Under the generalized central limit theorem assuming independence, GMPs provide powerful thresholds for r \leq -1, outperforming conservative approaches like robust risk analysis in simulations with Wishart or multivariate gamma dependence structures, while maintaining control over false positives. In probabilistic inference assessment, the generalized mean evaluates model calibration by computing the geometric mean of reported probabilities (for accuracy), arithmetic mean (for decisiveness), and a robust mean with exponent -2/3 (for stability against errors), allowing visualization of over- or under-confidence via metric angles in probability histograms. These applications underscore the versatility of generalized means in enhancing statistical inference and distribution characterization without relying on strict moment existence.^[35]^[36]

Economics and Optimization

In economics, generalized means, particularly power means, play a central role in modeling production functions through the constant elasticity of substitution (CES) framework, which captures the degree of substitutability between inputs such as capital and labor. The CES production function is formulated as Q = \gamma \left( \delta K^{\rho} + (1 - \delta) L^{\rho} \right)^{1/\rho}, where \rho = 1 - \sigma and \sigma denotes the elasticity of substitution; this structure aggregates weighted inputs via a power mean of order \rho, allowing flexibility in how factors combine, from perfect substitutes (\rho \to 1) to complements (\rho \to -\infty). Introduced by Arrow et al. in their seminal analysis of cross-country data, this form demonstrated that substitution elasticities vary across economies, challenging the fixed-proportions assumption of earlier models and influencing empirical studies of technological change and growth.^[37] Generalized means also underpin key inequality measures in income distribution analysis, where ratios between power means of different orders quantify disparity. For example, the ratio of the arithmetic mean (order p=1) to the geometric mean (order p=0) serves as a scale-invariant indicator of inequality, as the arithmetic-geometric mean inequality implies this ratio exceeds 1, with equality holding only when all incomes are identical; this ratio has been applied to assess dispersion in wage and wealth data. Atkinson's inequality index extends this by incorporating an aversion parameter \epsilon > 0, defining the equally distributed equivalent income as a power mean of order $1 - \epsilon and the index as $1 minus the ratio of this equivalent to the total mean income, thus weighting lower incomes more heavily for progressive evaluations. This measure, derived axiomatically, has shaped welfare economics by linking inequality assessment to social welfare functions.^[38] In optimization contexts, power means provide objective functions for resource allocation problems, balancing efficiency and equity by minimizing a power mean of losses or deviations. For instance, in fair division or welfare maximization, the goal may be to minimize the p-norm (a power mean of order p) of agents' disutilities from allocated resources, where low p (e.g., p=1) emphasizes absolute fairness and high p (e.g., p \to \infty) prioritizes avoiding extreme inequities; this approach ensures homogeneity and scalability in economic models. Such formulations appear in operations research applications to economic planning, like distributing public goods, where the power mean objective aligns with axioms of symmetry and monotonicity.^[39]

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### Section on Power Means Inequalities
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