On subadditive quasi-arithmetic means

Abstract

Let $f:{\mathbb{R}}_{+}\to \mathbb{R}$ 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 ${\mathcal{A}}_f:{\mathbb{R}}_{+}^n\to {\mathbb{R}}_{+}$ 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 $\hat{I}\pm \in [0,\infty ]$ such that $f_{+}^{\prime \prime }:=(f^{\prime }{)}_{+}^{\prime }$ is positive on $(0,\hat{I}\pm )$ and $f_{+}^{\prime \prime }=0$ on $[\hat{I}\pm ,\infty )$, furthermore, $\frac{f^{\prime }}{f_{+}^{\prime \prime }}$ is increasing and superadditive on $(0,\hat{I}\pm )$.