On subadditive quasi-arithmetic means

Let $f\text{\hspace{0.17em}⁣}:{\mathbb{R}}_{+}\to \mathbb{R}$ be a continuous and strictly monotone function. In the main result of this paper, we show that, for a fixed $n\ge 2$, the $n$-variable mean ${\mathcal{A}}_f\text{\hspace{0.17em}⁣}:{\mathbb{R}}_{+}^n\to {\mathbb{R}}_{+}$ defined by $${\mathcal{A}}_f(x_1,\dots ,x_n):=f^{-1}\left(\frac{f(x_1)+\cdots +f(x_n)}{n}\right)$$ is subadditive if and only if $f$ is differentiable with a continuously semi-differentiable and nonvanishing first derivative, and there exists an $\alpha \in [0,\mathrm{\infty }]$ such that $f_{+}^{\mathrm{\prime }\mathrm{\prime }}:=(f^{\mathrm{\prime }}{)}_{+}^{\mathrm{\prime }}$ is positive on $(0,\alpha )$ and $f_{+}^{\mathrm{\prime }\mathrm{\prime }}=0$ on $[\alpha ,\mathrm{\infty })$, furthermore, $\frac{f^{\mathrm{\prime }}}{f_{+}^{\mathrm{\prime }\mathrm{\prime }}}$ is increasing and superadditive on $(0,\alpha )$.