On a lattice-like property of quasi-arithmetic means.

We will prove that every pair of quasi-arithmetic means satisfying certain smoothness assumption has the supremum and the infimum in this set. More precisely, if f and g are C 2 functions with nowhere vanishing first derivative then there exists a function h such that: (i) A [ f ] ≤ A [ h ] , (ii) A [ g ] ≤ A [ h ] , and (iii) for every continuous strictly monotone function s : I → R A [ f ] ≤ A [ s ] and A [ g ] ≤ A [ s ] implies A [ h ] ≤ A [ s ] (A [ f ] stands for a quasi-arithmetic mean). Moreover, h ∈ C 2 , h ′ ≠ 0 , and it is a solution of the differential equation h ″ h ′ = max ⁡ ( f ″ f ′ , g ″ g ′ ). We also provide some extension to a finite family of means and dual (reflected) result. [ABSTRACT FROM AUTHOR]