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Multivariable generalizations of bivariate means via invariance

Paweł Pasteczka

Abstract

For a given $p$-variable mean $M \colon I^p \to I$ ($I$ is a subinterval of $\mathbb{R}$), following (Horwitz, 2002) and (Lawson and Lim, 2008), we can define (under certain assumption) its $(p+1)$-variable $β$-invariant extension as the unique solution $K \colon I^{p+1} \to I$ of the functional equation \begin{align*} K\big(M(x_2,\dots,x_{p+1})&,M(x_1,x_3,\dots,x_{p+1}),\dots,M(x_1,\dots,x_p)\big)\\ &=K(x_1,\dots,x_{p+1}), \text{ for all }x_1,\dots,x_{p+1} \in I \end{align*} in the family of means. Applying this procedure iteratively we can obtain a mean which is defined for vectors of arbitrary lengths starting from the bivariate one. The aim of this paper is to study the properties of such extensions.

Multivariable generalizations of bivariate means via invariance

Abstract

For a given -variable mean ( is a subinterval of ), following (Horwitz, 2002) and (Lawson and Lim, 2008), we can define (under certain assumption) its -variable -invariant extension as the unique solution of the functional equation \begin{align*} K\big(M(x_2,\dots,x_{p+1})&,M(x_1,x_3,\dots,x_{p+1}),\dots,M(x_1,\dots,x_p)\big)\\ &=K(x_1,\dots,x_{p+1}), \text{ for all }x_1,\dots,x_{p+1} \in I \end{align*} in the family of means. Applying this procedure iteratively we can obtain a mean which is defined for vectors of arbitrary lengths starting from the bivariate one. The aim of this paper is to study the properties of such extensions.
Paper Structure (14 sections, 17 theorems, 49 equations)

This paper contains 14 sections, 17 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. General framework
  3. Properties of means on the interval
  4. Invariance in a family of means
  5. Extended means
  6. Invariance of extended means
  7. Results
  8. Auxiliary results
  9. Main result
  10. Multivariable generalization of bivariate means
  11. Conjugacy of means
  12. Applications to classical families of means
  13. Quasiarithmetic envelopes
  14. Gini means

Key Result

Proposition 1.1

Assume that $\mu \colon X^k \to X$ is a topological k-mean and that the corresponding barycentric operator $\beta_\mu$ is power convergent. Define $\widetilde{\mu} : X^{k+1} \to X$ by $\widetilde{\mu}(x) = x^*$, where $\lim_{n\to\infty} \beta_\mu^n(x)=(x^*,\dots,x^*)$.

Theorems & Definitions (28)

  • Proposition 1.1: LawLim08, Proposition 2.4
  • Proposition 2.1: Pas23b, Theorem 2
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 1
  • proof
  • Proposition 3.3
  • proof
  • ...and 18 more