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 $s:I\to \mathbb{R}$ $$A^{[f]}\le A^{[s]}\text{ and }{A}^{[g]}\le A^{[s]}\text{ implies }{A}^{[h]}\le A^{[s]}$$ ($A^{[f]}$ stands for a quasi-arithmetic mean generated by a function $f$ and so on). Moreover $h\in {\mathcal{C}}^2$, $h^{\prime }\ne 0$, and it is a solution of the differential equation $$\frac{h^{\prime \prime }}{h^{\prime }}=\max (\frac{f^{\prime \prime }}{f^{\prime }},\frac{g^{\prime \prime }}{g^{\prime }}).$$ 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.

References in corpus (2)

• On Kedlaya type inequalities for weighted means
• Lower estimation of the difference among quasi-arithmetic means

Cited by in corpus (1)

• On the Jensen convex and Jensen concave envelopes of means