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On subadditive quasi-arithmetic means

Zsolt Páles, Paweł Pasteczka

Abstract

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 $(0,α)$.

On subadditive quasi-arithmetic means

Abstract

Let be a continuous and strictly monotone function. In the main result of this paper, we show that, for a fixed , the -variable mean defined by is subadditive if and only if is differentiable with a continuously semi-differentiable and nonvanishing first derivative, and there exists an such that is positive on and on , furthermore, is increasing and superadditive on .
Paper Structure (4 sections, 8 theorems, 71 equations)

This paper contains 4 sections, 8 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Semi-differentiability, Monotonicity and Convexity
  3. A Necessary Condition for the Subadditivity of General Means
  4. Subadditivity of Quasi-arithmetic Means

Key Result

Theorem A

Let $n\geq2$ be fixed and $f \colon I \to \mathbb{R}$ be continuous and strictly monotone. Then the $n$-variable mean $\mathscr{A}_f|_{I^n}$ is Jensen convex if and only if $f$ is twice continuously differentiable with a nonvanishing first derivative and either $f"$ is identically zero on $I$, or $f

Theorems & Definitions (13)

  • Theorem A
  • Lemma
  • proof
  • Lemma
  • Lemma
  • Lemma
  • proof
  • Lemma
  • proof
  • Theorem
  • ...and 3 more