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

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.