Limit properties in a family of quasi-arithmetic means

TL;DR

This paper investigates the limit behavior of quasi-arithmetic means, providing conditions under which they exhibit properties similar to power means as arguments tend to infinity.

Contribution

It establishes necessary and sufficient conditions for quasi-arithmetic means to have limit properties analogous to power means under certain smoothness assumptions.

Findings

Identifies conditions for quasi-arithmetic means to converge pointwise to a maximum

Extends known properties of power means to a broader class of means

Provides a theoretical framework for understanding limit behaviors of quasi-arithmetic means

Abstract

It is known that the family of power means tends to maximum pointwise if we pass argument to infinity. We will give some necessary and sufficient condition for the family of quasi-arithmetic means generated by a functions satisfying certain smoothness conditions to have analogous property.