On subadditive quasi-arithmetic means

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