On the new smoothness class of means and its impact to mean-type mappings

TL;DR

This work introduces and elaborates the notion of residual means, capturing near-diagonal behavior of symmetric means through a residuum function $\xi_M$. It proves that every symmetric locally $\mathcal{C}^3$ mean is residual and derives explicit residuums for major mean families (quasideviation, Bajraktarevi\'c, Gini, and quasiarithmetic), including a practical limit formula for variance ratios in mean-type iterations built from residual means. The results extend to means on variable numbers of arguments and reveal independence of the residuum from arity under repetition invariance. A key outcome is a general limit for $\displaystyle \lim_{n\to\infty} \frac{\mathrm{Var}\ \mathbf{M}^{n+1}(x)}{(\mathrm{Var}\ \mathbf{M}^n(x))^2}$ expressed via the residua evaluated at the invariant mean, enabling quadratic-type convergence analyses for invariant mean mappings and concrete applications to families like Gini and Bajraktarevi\'c means.

Abstract

We define so-called residual means, which have a Taylor expansion of the form $M(x)=\bar x +\tfrac12 ξ_M(\bar x) \text{Var}(x)+o(\|x-\bar x\|^α)$ for some $α>2$ and a single-variable function $ξ_M$ ($\bar x$ stands for the arithmetic mean of the vector $x$), and show that all symmetric means which are three times continuously differentiable are residual. We also calculate the value of residuum for quasideviation means and a few subclasses of this family. Later, we apply it to establish the limit of the sequence $\big(\frac{\text{Var}\ {\bf M}^{n+1}(x)}{(\text{Var}\ {\bf M}^n(x))^2}\big)_{n=1}^\infty$, where ${\bf M} \colon I^p\to I^p$ is a mean-type mapping consisting of $p$-variable residual means on an interval $I$, and $x \in I^p$ is a nonconstant vector.

Theorems & Definitions (25)

• Theorem A
• Theorem B: BorBor87, Theorem 8.8
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Remark 1
• Remark 2
• Theorem 2.3
• proof