Extremal Means in Analysis and Combinatorics

• Extremal means are defined as the maximal or minimal values of mean-type functionals under specific structural, distributional, or combinatorial constraints.
• Analysis methods reduce complex configurations to one-dimensional problems using monotonicity, variational techniques, and critical-point analysis to derive sharp bounds.
• Applications span uniform distribution theory, optimal transport, multivariate extreme value analysis, spherical clustering, and combinatorial pattern avoidance.

Extremal means describe the maximal or minimal values, or configurations, of mean-type functionals under specific structural, distributional, or combinatorial constraints. These functionals appear throughout probability, functional analysis, combinatorics, uniform distribution theory, and multivariate statistics, serving as canonical objects in optimization, extremal combinatorics, descriptive statistics of extremes, and the theory of inequalities. Extremal means may refer to optimizing classical means under restricted classes (e.g., copulas), means of analytic functions with extremal regularity or growth, or statistical prototypes capturing joint extremes. This article synthesizes key contexts, methodologies, and results for extremal means, referencing recent developments in uniform distribution theory, mean inequalities, spherical clustering for extremes, pattern avoidance in words, and analytic function theory.

1. Extremal Means and Best-Constant Inequalities for Power Means

Consider, for $n$ positive variables $x_1,\dots,x_n$, the ratio

$$R_{p,n}(x_1,\dots,x_n) = \frac{A_n-G_n}{M_{n,p}-G_n}$$

where

$$A_n = \frac{1}{n}\sum_{i=1}^n x_i,\quad G_n = \left(\prod_{i=1}^n x_i\right)^{1/n},\quad M_{n,p} = \left(\frac{1}{n}\sum_{i=1}^n x_i^p\right)^{1/p},\; p \neq 0,1.$$

The extremal problem seeks sharp bounds for $R_{p,n}$ over all configurations $(x_1,\dots,x_n)$ subject to normalization $\sum x_i=1$ (Aliyev, 20 May 2024). It is proven that extrema for $R_{p,n}$ are attained among two-point configurations; this reduces the $n$-variable problem to piecewise analysis of a single-variable function $q(x)$ on $[0,1/(n-1)]$: $q(x) = \frac{G_n(x)-A_n}{M_{n,p}(x)-A_n}$ Monotonicity and critical-point analysis reveal four regimes, depending on $p$ and $n$, yielding endpoint and (possibly) interior extremal values. Explicit best constants in inequalities interpolating $A_n$, $G_n$, and $M_{n,p}$ are constructed, such as

$$\left(\frac{n}{n-1}\right)^{p-1} M_{n,p} + \left[1-\left(\frac{n}{n-1}\right)^{p-1}\right] G_n \leq A_n \leq n^{p-1} M_{n,p} + [1-n^{p-1}] G_n$$

for suitable $p,n$ ranges. The boundary behavior and monotonicity in $p$ and $n$ are fully characterized. This structural reduction underlies general best-constant results for interpolated means and highlights the geometric simplicity of extremal mean configurations.

2. Extremal Problems in Uniform Distribution Theory and Optimal Transport

In uniform distribution and copula theory, the extremal mean problem is to optimize functionals

$I(F) = \int_0^1\int_0^1 F(x,y)\,dC(x,y)$

over all bivariate distribution functions (copulas $C$) with uniform marginals (Baláž et al., 2015). The value of $I(F)$ can be interpreted as a limiting mean for uniformly distributed sequences or as an optimal transport cost. The extremal copula depends only on the sign structure of the mixed partial derivative $D_2(x,y) = \partial^2 F(x,y)/\partial x \partial y$. For $D_2 > 0$, $C^* = \min(x,y)$ achieves the supremum; for $D_2 < 0$, $C^* = \max(x + y - 1, 0)$ is extremal.

For functions $F$ where $D_2$ changes sign, the optimizer is constructed piecewise using min/max-type formulas glued according to sign domains. The general optimization problem reduces to a one-dimensional variational system with appropriate copula side-constraints. In examples, piecewise linear or trigonometric costs yield explicit extremal means via solution of Euler-Lagrange ODEs or optimal coupling strategies. This framework connects extremal means to the Monge–Kantorovich optimal transport problem, leveraging c-convexity and copula theory to compute maximal and minimal integrals for arbitrary $F$.

3. Extremal Area Integral Means in Weighted Bergman Spaces

In the context of analytic function spaces on the unit disc $D$, extremal means concern the growth rate of weighted area integral means

$$M_{p,\alpha}(r,f) = \left( \int_{|z|<r} |f(z)|^p (1 - |z|^2)^\alpha\,dA(z) / \int_{|z|<r} (1 - |z|^2)^\alpha\,dA(z) \right)^{1/p}$$

for $f \in A^p_\alpha$ (the weighted Bergman space), $1 \leq p < \infty$, and weight exponent $-1 < \alpha < \infty$ (Ferguson, 2016). Ferguson's results establish an equivalence: the extremal growth rate $$M_{p,\alpha}(r,f) = O((1-r)^{\beta - (1+\alpha)/p})$$ is achieved if and only if $f$ is mean Hölder continuous of order $\beta$ in the sense

$\|\Delta^2_h f\|_{A^p_\alpha} \le C |h|^\beta$

where $\Delta^2_h f$ denotes the second iterated difference. The regularity equivalence is proved with explicit bounds.

In weighted Bergman space extremal problems, one considers the maximization

$$\sup_{F \in A^p_\alpha, \|F\|=1} \operatorname{Re} \int_D F \overline{k} dA_\alpha$$

for given data $k \in A^q_\alpha$. Ferguson establishes that regularity of $k$, in terms of its mean Hölder order, is quantitatively inherited by the (unique) extremal solution $F$ with precise exponent loss, depending on $p$ and $\alpha$. This structure extends classical Hardy and Bergman space regularity transfer theorems to the full weighted case, with implications for boundary Hölder continuity and analytic extension.

4. Extremal Means in Multivariate Extreme Value Theory and Clustering

Clustering extremal observations in multivariate datasets requires identifying representational "prototypes" or mean directions that characterize the geometry of joint extremes (Janßen et al., 2019). In spherical $k$-means clustering of extremes, the framework involves:

• Preprocessing samples via marginal standardization and radial thresholding to extract high-norm observations.
• Projecting extreme points onto the unit sphere, forming angular components.
• Applying spherical $k$-means (minimizing angular error) among these angular observations to estimate the modes of the limiting spectral measure $S$ for joint extremes.

For empirical spectral measures $\hat S_n$ converging weakly to $S$, the spherical $k$-means cluster centers are shown to be consistent estimators of the extremal prototypes (Hausdorff convergence under mild regularity). In max-linear models, the method not only identifies discrete spectral atoms but also consistently estimates the associated weights and directions, providing a non-parametric alternative to likelihood-based inference.

Practical examples in environmental and financial data demonstrate that extremal means extracted via this methodology reveal the structure and timing of extreme-risk regimes, clarifying which groups of variables tend to exhibit simultaneous extremes.

5. Extremal Means in Combinatorics: Pattern-Avoidance

In combinatorics, extremal means are exemplified by binary words over $\Sigma_2 = \{0,1\}$ that are overlap-free or $\beta$-free and cannot be extended without violating the avoidance property (Mol et al., 2020). For a word $w$, being extremal overlap-free means that all its single-letter extensions admit an overlap, while $w$ itself remains overlap-free. Analogously, for $\beta$-free, extensions incur a prohibited exponent.

Mol, Rampersad, and Shallit provide a complete description of all binary word lengths $n$ for which an extremal overlap-free word exists. The characterization is given by the set

$$N = \{10, 12\} \cup \{2k : k \geq 10\} \cup \{2^k + 1 : k \geq 5\} \cup \{3 \cdot 2^k + 1 : k \geq 3\}$$

Arbitrarily long extremal $\beta$-free binary words exist for $2^+ \leq \beta \leq 8/3$, constructing infinite families that remain $\beta$-free but turn non-$\beta$-free upon any extension. Extremal words thus demarcate boundaries of pattern-avoidance, with their structure rooted in combinatorial rigidity and recursivity (Thue–Morse morphisms for word construction).

6. Synthesis and Theoretical Implications

Extremal means, across analytic function spaces, combinatorial configurations, multivariate extremes, and inequality theory, exhibit recurring themes: reduction to low-dimensional or structured boundary problems, inheritance of extremality from local sign or structural regimes, and connection to optimal transport or regularity theory. Frequently, extremal configurations are dictated by sign conditions, convexity, or symmetry, permitting sharp or, in some cases, unique characterization of extremal objects. Theoretical advances in identifying extremal means foster progress in best-constant inequalities, optimal estimation in high-dimensional statistics, and the analysis of maximal pattern-avoidance sets, establishing extremal means as a unifying concept in contemporary mathematical analysis and probability.