Lattice-like property of quasi-arithmetic means: revisited

TL;DR

This paper revisits the lattice-like structure of quasi-arithmetic means, demonstrating that certain families have optimal bounds within the same family, generated by functions with specific smoothness and boundedness properties.

Contribution

It establishes the existence of best bounds within families of quasi-arithmetic means generated by smooth functions with nonvanishing derivatives, extending previous understanding of their lattice structure.

Findings

Families of quasi-arithmetic means have well-defined best bounds within the same family.

These bounds are generated by functions with similar smoothness and boundedness properties.

The results reinforce the lattice-like organization of quasi-arithmetic means.

Abstract

We show that every family of quasi-arithmetic means generated by (a subset of) $\mathcal{C}^1$ functions with nonvanishing derivative which is bounded (from below or from above) by a quasi-arithmetic mean, possesses the best (lower or upper) bound which is a quasi-arithmetic mean generated by a function belonging to the same family.