On subadditive quasi-arithmetic means
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914398160814080 |
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| author | Páles, Zsolt Pasteczka, Paweł |
| author_facet | Páles, Zsolt Pasteczka, Paweł |
| contents | 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 $α\in[0,\infty]$ such that $f''_+:=(f')'_+$ is positive on $(0,α)$ and $f''_+=0$ on $[α,\infty)$, furthermore, $\frac{f'}{f''_+}$ is increasing and superadditive on $(0,α)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_15324 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On subadditive quasi-arithmetic means Páles, Zsolt Pasteczka, Paweł Classical Analysis and ODEs 26D15, 26E60, 39B62 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 $α\in[0,\infty]$ such that $f''_+:=(f')'_+$ is positive on $(0,α)$ and $f''_+=0$ on $[α,\infty)$, furthermore, $\frac{f'}{f''_+}$ is increasing and superadditive on $(0,α)$. |
| title | On subadditive quasi-arithmetic means |
| topic | Classical Analysis and ODEs 26D15, 26E60, 39B62 |
| url | https://arxiv.org/abs/2603.15324 |