Convexity and Concavity of $$\boldsymbol f$$-Potentials (Kolmogorov Means)

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