Convexity and concavity of $f$-potentials (Kolmogorov means)

TL;DR

This paper establishes criteria for the convexity and concavity of $f$-potentials, encompassing various means like arithmetic, geometric, harmonic, and $L^p$ norms, and explicitly characterizes functions $f$ that meet these criteria.

Contribution

It provides new theoretical criteria for convexity and concavity of $f$-potentials and explicitly characterizes all functions $f$ satisfying these conditions.

Findings

Criteria for convexity and concavity of $f$-potentials are established.

All functions $f$ satisfying these criteria are explicitly computed.

Includes special cases like arithmetic, geometric, harmonic means, and $L^p$-norms.

Abstract

In the paper we prove criteria for convexity and concavity of $f$-potentials ($f$-means, Kolmogorov means, weighted quasi-arithmetic means), which particular cases are the arithmetic, geometric, harmonic means, the thermodynamic potential (exponential mean), and the $L^{p}$-norm. Then we compute in quadratures all functions $f$ satisfying these criteria.