When is a family of generalized means a scale?

TL;DR

This paper characterizes when a family of generalized means forms a scale, providing conditions under which a parameterized family of functions generates a continuous bijection, thus defining a scale on an interval.

Contribution

The paper establishes precise conditions under which a family of functions generates a scale, offering new proofs and extending classical results in the theory of generalized means.

Findings

The family {k_t} generates a scale if certain monotonicity and regularity conditions are met.

The mapping from parameter t to the inverse of the mean is a continuous bijection.

The results include new proofs of classical theorems in the theory of means.

Abstract

For a family {k_t | t \in I} of real C^2 functions defined on U (I, U -- open intervals) and satisfying some mild regularity conditions, we prove that the mapping I \ni t --> k_t^{-1}(\sum_{i=1}^n w_i k_t(a_i)) is a continuous bijection between I and (min a, max a), for every fixed non-constant sequence a = (a_i)_{i=1}^n with values in U and every set, of the same cardinality, of positive weights w=(w_i)_{i=1}^n. In such a situation one says that the family of functions {k_t} generates a scale on U. The precise assumptions in our result read (all indicated derivatives are with respect to x \in U) (i) k'_t does not vanish anywhere in U for every t \in I, (ii) I \ni t \mapsto \frac{k"_t(x)}{k'_t(x)} is increasing, 1--1 on a dense subset of U and onto the image R for every x \in U. This result makes possible few new things as well as new proofs of classical results.