On a lattice-like property of quasi-arithmetic means

We will prove that in a family of quasi-arithmetic means sattisfying certain smoothness assumption (embed with a naural pointwise ordering) every finite family has both supremum and infimum, which is also a quasi-arithmetic mean sattisfying the same smoothness assumptions. More precisely, if $f$ and $g$ are ${\mathcal{C}}^2$ functions with nowhere vanishing first derivative then there exists a function $h$ such that: (i) $A^{[f]}\le A^{[h]}$, (ii) $A^{[g]} \le A^{[h]}$,and(iii)foreverycontinuousstrictlymonotonefunction$s \colon I \to $\mathbb{R}$$ $$A^{[f]}\le A^{[s]}\text{ and }{A}^{[g]}\le A^{[s]}\text{ implies }{A}^{[h]}\le A^{[s]}$$ ($A^{[f]}$ stands for a quasi-arithmetic mean generated by a function $f$ and so on). Moreover $h\in {\mathcal{C}}^2$, $h^{\mathrm{\prime }}\mathrm{\ne }0$, and it is a solution of the differential equation $$\frac{h^{\mathrm{\prime }\mathrm{\prime }}}{h^{\mathrm{\prime }}}=\max (\frac{f^{\mathrm{\prime }\mathrm{\prime }}}{f^{\mathrm{\prime }}},\frac{g^{\mathrm{\prime }\mathrm{\prime }}}{g^{\mathrm{\prime }}})\mathrm{.}$$ We also provide some extension to a finite family of means. Obviously dual statements with inverses inequality sign as well as a multifuntion generalization will be also stated.