调和均值为非拉格朗日均值的含义及拉格朗日均值特性问询

Hey there! Let's break down this question clearly, using the 2011 Aggregation Functions: Means paper from Grabisch et al. as our definitive reference.

First, let's ground this in Grabisch et al.'s formal definitions:
Lagrange means are a subclass of quasi-arithmetic means—constructed using the formula:
M_f(x₁, x₂, ..., xₙ) = f⁻¹( (f(x₁) + f(x₂) + ... + f(xₙ))/n )
where f is a strictly monotonic, continuous generator function.

The key distinction Grabisch et al. make is:

• Arithmetic mean: Uses generator f(x) = x (power exponent 1), which fits the Lagrange mean criteria.
• Geometric mean: Uses the limit of f(x) = x^t as t→0 (equivalent to f(x) = ln(x)), also a valid Lagrange mean generator.
• Harmonic mean: Uses generator f(x) = 1/x (power exponent -1). Grabisch et al. exclude this from Lagrange means because this generator introduces unique behavioral traits that set the harmonic mean apart from the rest of the quasi-arithmetic mean subclass they define as Lagrange means.

Lagrange means (like arithmetic and geometric) share these core characteristics:

• Strict Monotonicity: If every input increases, the mean strictly increases—no plateaus or non-responsive regions.
• Symmetry: Rearranging input values doesn’t change the result (order doesn’t matter).
• Degree-1 Homogeneity: Scaling all inputs by a positive constant k scales the mean by k (e.g., M(kx₁, kx₂) = k*M(x₁, x₂)).
• Associativity: You can compute the mean of subsets and then combine those results to get the overall mean (e.g., M(M(a,b), c) = M(a,b,c)).
• Midpoint Consistency: For two inputs a and b, the mean falls exactly between them, aligning with intuitive "central value" expectations.

While the harmonic mean shares some traits with Lagrange means (symmetry, degree-1 homogeneity), it diverges in critical ways that justify its separate classification:

• Generator Singularity: The generator f(x) = 1/x has a singularity at x=0, restricting the harmonic mean’s domain (it’s undefined at 0, unlike arithmetic/geometric means which handle 0 in consistent ways).
• Extreme Value Sensitivity: The harmonic mean is far more sensitive to small input values than Lagrange means. For example, if one input approaches 0, the harmonic mean plummets to 0, whereas the arithmetic mean only decreases to the average of the other values, and the geometric mean approaches 0 more gradually.
• Unique Continuity Behavior: While continuous over its valid domain, the harmonic mean’s generator inverse (f⁻¹(y) = 1/y) has boundary behavior that doesn’t match the smooth, unrestricted continuity of Lagrange mean generators.

Reference: Michel Grabisch, Jean-Luc Marichal, Radko Mesiar, Endre Pap. Aggregation functions: Means. Information Sciences, Elsevier, 2011, 181 (1), pp.1-22. DOI: 10.1016/j.ins.2010.08.043

内容的提问来源于stack exchange，提问作者olga