Reducible means

Abstract

A n variables mean M is said to be reducible in a certain class of means $\mathcal {N}$ when M can be represented as a composition of a finite number $M_{0},\ldots ,M_{r}$ of means belonging to $\mathcal {N}$, being less than n the number of variables of every $M_{i}$. In this paper, a basic classification of reducible means is developed and the notions of S-reducibility, a type of analytically decidible reducibility, and of complete reducibility of a mean are isolated. Several applications of these notions are presented. In particular, a continuous and scale invariant weighting procedure defined on a class $\mathcal {M}_{2}$ of two variables means is extended without losing its properties to the class of reducible means in $\mathcal {M}_{2}$.

Appendix

To facilitate the reading of the paper, a table of the main notations employed is given below.

$\mathbb {R}^{+}$ :

Set of positive real numbers

I :

Real interval

$w^{T}$ :

Transpose of w

$\mathcal {\preceq }$ :

Product order in $I^{n}$

$\nu \left( F\right)$ :

Number of effective arguments of the function F

$S_{n}$ :

Symmetric group of order n

$X_{k}\left( x_{1},\ldots ,x_{n}\right) \equiv x_{i},\ k=1,\ldots ,n$,:

Coordinate means

$X_{n}^{( k) },$$k=1,\ldots ,n$,:

Order means

$\min _{n},\ \max _{n}$ :

n variables extremal means

$A_{n}$ :

n variables arithmetic mean

$G_{n}$ :

n variables geometric mean

$L_{n}$ [$L_{n,\left( w_{1},\ldots ,w_{n}\right) }$]:

n variables linear mean, [with weights $\left( w_{1},\ldots ,w_{n}\right)$]

$G_{n}$[$G_{n,\left( w_{1},\ldots ,w_{n}\right) }$:

n variables geometric mean [with weights $\left( w_{1},\ldots ,w_{n}\right)$]

$QL_{n}$ [$QL_{n.\left( w_{1},\ldots ,w_{n}\right) }$:

n variables quasilinear mean [with weights $\left( w_{1},\ldots ,w_{n}\right)$]

$FH_{n}\left( x_{1},\ldots ,x_{n}\right)$ :

The n variables mean defined by $\left( \sum \nolimits _{i=1}^{n}f_{i}\left( x_{i}\right) x_{i}\right) /\sum \nolimits _{i=1}^{n}f_{i}\left( x_{i}\right)$

$CH_{n}^{( r) }\qquad r$th:

Weighted n variables counter-harmonic mean

$e_{n}^{[ r] }\left( x_{1},\ldots ,x_{n}\right)$ :

The rth symmetric polynomial function

$\mathfrak {S}_{n}^{[ r] }\left( x_{1},\ldots ,x_{n}\right)$ :

The rth symmetric polynomial mean

$P_{\alpha }^{( r) }$ :

The (two variables) weighted power mean with exponent r and weight $\alpha$

$\mathcal {M}( i) ,\,\mathcal {N}( i)$ :

General classes of means defined on I

$\left( \mathcal {M}\right) _{f}\left( f( i) \right)$ :

The class of means $\left\{ \left( M_{f}\right) :M\in \mathcal {M}( i) \right\}$

$\mathcal {C}^{( k) }\mathcal {M}_{n}( i)$ :

The class of n variables $\mathcal {C}^{( k) }$ means defined on I

$\mathcal {LM}( i)$ :

The class of linear means defined on I

$\mathcal {LM}_{\mathbb {Q}}\left( \mathbb {R}\right)$ :

The class of linear means with a rational weight vector

$\mathcal {QLM}( i)$ :

The class of quasilinear means defined on I

$\mathcal {QLM}_{2,\mathbb {Q}}\left( \mathbb {R}^{+}\right)$ :

The class of two variables quasilinear means with rational weights defined on $\mathbb { R}^{+}$

$\mathfrak {S}\mathcal {M}\left( \mathbb {R}^{+}\right)$ :

The class of polynomial symmetric means

$\left( M\right) _{f}$ :

The mean conjugate of M by the homeomorphism f

$\overline{M}$ :

Two variables mean complementary of M

$\mathbf {n}$ :

The set (of indices) $\left\{ 1,\ldots ,n\right\}$

J :

a subset of $\mathbf {n}$

[J]:

The increasingly ordered set of indices corresponding to $J\subseteq \mathbf{n}$

$\left( x_{j}\right) _{\left[ J\right] }$ :

The k-tuple $( x_{i_{1}},\ldots ,x_{i_{k}})$ with $\left[ i_{1},\ldots ,i_{k}\right] =\left[ J\right]$ ($J=\left\{ i_{1},\ldots ,i_{k}\right\} \subseteq \mathbf { n}$)

$M_{[ i_{1},\ldots ,i_{k}] }\left( x_{i_{1}},\ldots ,x_{i_{k}};u\right)$ :

The specialization of variables of M obtained by setting $x_{j}=u,\ j\notin \left\{ i_{1},\ldots ,i_{k}\right\}$

$M_{J}\left( \left( x_{j}\right) _{\left[ \varvec{n}\setminus J\right] };u\right)$ :

The specialization of variables of M obtained by setting $x_{j}=u,\ j\in J$

$M_{J_{1}\cdots J_{r}}\left( \left( x_{j}\right) _{\left[ n\setminus \cup _{i}J_{i}\right] };u_{1},\ldots ,u_{r}\right)$ :

The specialization of variables of M obtained by setting $x_{j}=u_{i},\ j\in J_{i},\ i=1,\ldots ,r$

$\mu _{[ i_{1},\ldots ,i_{k}] }\qquad \left[ i_{1},\ldots ,i_{k} \right]$ :

-lower mean of a mean M

$\mathcal {FS}( \mathfrak {F})$ :

The set of functional symbols in a formula $\mathfrak {F}$

$\mathcal {VAR}( \mathfrak {F})$ :

The set of variables in a formula $\mathfrak {F}$

$T( \mathfrak {F})$ :

The tree of the formula F

$V\left( G\right)$ :

Set of vertices of a graph G

$\hbox {v}\,\left( T\right)$ :

Order (number of vertices) of a tree T

$\hbox {a}\,\left( T\right)$ :

Size (number of arcs) of a tree T

$\hbox {root} \; \left( T\right)$ :

Root vertex of a tree T

$\hbox {nl}\,\left( T\right)$ :

The number of leaves of a tree T

$\hbox {h}\,\left( T\right)$ :

The height of a tree T (the length of the longest path joining the root with a leaf)

$\hbox {des}\left( v\right)$ :

The descent of a vertex $v\in V\left( T\right)$ (the number of subtrees of v)

$\mathcal {W}:\mathcal {M}_{n}(i) \times \Delta _{n-1}\rightarrow \mathcal {N}( i)$ :

Weighting procedure defined on the class of n variables means $\mathcal {M}_{n}( i)$

$\mathcal {A}\left( M;I\right)$ :

The family of M-affine functions

$\mathcal {BA}\left( M;I\right)$ :

The family of bijective M-affine functions