Title: Interval-type theorems concerning quasi-arithmetic means,
Abstract: Family of quasi-arithmetic means has a natural, partial order (point-wise
order) $A^{[f]}\le A^{[g]}$ if and only if $A^{[f]}(v)\le A^{[g]}(v)$ for all
admissible vectors $v$ ($f,\,g$ and, later, $h$ are continuous and monotone and
defined on a common interval). Therefore one can introduce the notion of
interval-type sets (sets $\mathcal{I}$ such that whenever $A^{[f]} \le A^{[h]}
\le A^{[g]}$ for some $A^{[f]},\,A^{[g]} \in \mathcal{I}$ then $A^{[h]} \in
\mathcal{I}$ too). Our aim is to give examples of interval-type sets involving
vary smoothness assumptions of generating functions.