2012/2013 When is a Family of Generalized Means a Scale?
Paweł Pasteczka
Real Anal. Exchange 38(1): 193-210 (2012/2013).

## Abstract

For a family $$\{k_\alpha \,\vert \,\alpha \in I\}$$ of real $$\mathcal{C}^2$$ functions defined on $$U$$ ($$I$$, $$U$$ — open intervals) and satisfying some mild regularity conditions, we prove that the mapping $$I \ni \alpha \mapsto k_\alpha^{-1}\bigl(\sum_{i=1}^n w_i k_\alpha(a_i)\bigr)$$ is a continuous bijection between $$I$$ and $$(\min\underline{a}, \,\max\underline{a})$$, for every fixed non-constant sequence $$\underline{a} = \bigl(a_i\bigr)_{i=1}^n$$ with values in $$U$$ and every set, of the same cardinality, of positive weights $$\underline{w} = \bigl(w_i\bigr)_{i=1}^n$$. In such a situation one says that the family of functions $$\{k_\alpha\}$$ generates a scale on $$U$$. The precise assumptions in our result read (all indicated derivatives are with respect to $$x \in U$$)

(i) $$k'_\alpha$$ vanishes nowhere in $$U$$ for every $$\alpha \in I$$,

(ii) $$I \ni \alpha \mapsto \frac{k''_\alpha(x)}{k'_\alpha(x)}$$ is increasing, 1-1 on a dense subset of $$U$$ and onto the image $$\mathbb{R}$$ for every $$x \in U$$.

This result makes possible three things: 1) a new and extremely short proof of the classical fact that power means generate a scale on $$(0,+\infty)$$, 2) a short proof of a fact, which is in a direct relation to two results established by Kolesárová in 2001, that, for every strictly increasing convex and $$\mathcal{C}^2$$ function $$k \colon (0,\,1) \to (0,\,+\infty)$$, the class $$\{\mathfrak{M}_{k_\alpha}\}_{\alpha \in (0,\,+\infty)}$$ of quasi-arithmetic means (see Introduction for the definition) generated by functions $$k_\alpha$$, $$k_\alpha(x) = k(x^\alpha)$$, $$\alpha \in (0,\,+\infty)$$, generates a scale on $$(0,1)$$ between the geometric mean and maximum (meaning that, for every $$\underline{a}$$, $$\underline{w}$$, if $$s \in \bigl(\prod_{i = 1}^n a_i^{\,w_i},\,\max(\underline{a})\bigr)$$ then there exists exactly one $$\alpha$$ such that $$\mathfrak{M}_{k_\alpha}(\underline{a},\underline{w}) = s$$). 3) a brief proof of one of the classical results of the Italian statistics' school from the 1910-20s that the so-called radical means generate a scale on $$(0,\, +\infty)$$.

## Citation

Paweł Pasteczka. "When is a Family of Generalized Means a Scale?." Real Anal. Exchange 38 (1) 193 - 210, 2012/2013.

## Information

Published: 2012/2013
First available in Project Euclid: 29 April 2013

zbMATH: 1277.26061
MathSciNet: MR3083206

Subjects:
Primary: 26E60
Secondary: 47A63 , 47A64

Keywords: generalized mean , Inequalities‎ , ‎mean‎ , ‎quasi-arithmetic mean , scale of means