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.
Categories: math.CA