On a lattice-like property of quasi-arithmetic means

Abstract: 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) for every continuous strictly monotone function $s \colon I \to \mathbb{R}$ $$ A^{[f]} \le A^{[s]} \text{ and } A^{[g]} \le A^{[s]} \text{ implies } A^{[h]} \le A^{[s]} $$ ($A^{[f]}$ stands for a quasi-arithmetic mean generated by a function $f$ and so on). Moreover $h\in\mathcal{C}^2$, $h'\ne0$, and it is a solution of the differential equation $$ \frac{h''}{h'}=\max\Big(\frac{f''}{f'},\,\frac{g''}{g'}\Big). $$ 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.