## Real Analysis Exchange

### When is a Family of Generalized Means a Scale?

Paweł Pasteczka

#### 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 $(0,\, +\infty)$.

#### Article information

Source
Real Anal. Exchange, Volume 38, Number 1 (2012), 193-210.

Dates
First available in Project Euclid: 29 April 2013

https://projecteuclid.org/euclid.rae/1367265648

Mathematical Reviews number (MathSciNet)
MR3083206

Zentralblatt MATH identifier
1277.26061

Subjects
Secondary: 47A63: Operator inequalities 47A64: Operator means, shorted operators, etc.

#### Citation

Pasteczka, Paweł. When is a Family of Generalized Means a Scale?. Real Anal. Exchange 38 (2012), no. 1, 193--210. https://projecteuclid.org/euclid.rae/1367265648

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