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

We define so-called residual means, which have a Taylor expansion of the form $M(x)=\bar{x}+\frac{1}{2}{\xi }_M(\bar{x})\text{Var}(x)+o(\mathrm{\parallel }x-\bar{x}{\mathrm{\parallel }}^{\alpha })$ for some $\alpha >2$ and a single-variable function ${\xi }_M$ ($\bar{x}$ 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 $(\frac{\text{Var }{\mathbf{M}}^{n+1}(x)}{(\text{Var }{\mathbf{M}}^n(x){)}^2}{)}_{n=1}^{\mathrm{\infty }}$, where $\mathbf{M}\text{\hspace{0.17em}⁣}:I^p\to I^p$ is a mean-type mapping consisting of $p$-variable residual means on an interval $I$, and $x\in I^p$ is a nonconstant vector.