Interval-type theorems concerning quasi-arithmetic means

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 $A^{[f]} \le A^{[h]} \le A^{[g]}$forsome$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.