# Subadditive functions and partial converses of Minkowski's and Mulholland's inequalities

Fundamenta Mathematicae (1993)

• Volume: 143, Issue: 1, page 75-85
• ISSN: 0016-2736

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## Abstract

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Let ϕ be an arbitrary bijection of ${ℝ}_{+}$. We prove that if the two-place function ${\varphi }^{-1}\left[\varphi \left(s\right)+\varphi \left(t\right)\right]$ is subadditive in ${ℝ}_{+}^{2}$ then $\varphi$ must be a convex homeomorphism of ${ℝ}_{+}$. This is a partial converse of Mulholland’s inequality. Some new properties of subadditive bijections of ${ℝ}_{+}$ are also given. We apply the above results to obtain several converses of Minkowski’s inequality.

## How to cite

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Matkowski, J., and Świątkowski, T.. "Subadditive functions and partial converses of Minkowski's and Mulholland's inequalities." Fundamenta Mathematicae 143.1 (1993): 75-85. <http://eudml.org/doc/211993>.

@article{Matkowski1993,
abstract = {Let ϕ be an arbitrary bijection of $ℝ_+$. We prove that if the two-place function $ϕ^\{-1\}[ϕ (s)+ϕ (t)]$ is subadditive in $ℝ^2_+$ then $ϕ$ must be a convex homeomorphism of $ℝ_+$. This is a partial converse of Mulholland’s inequality. Some new properties of subadditive bijections of $ℝ_+$ are also given. We apply the above results to obtain several converses of Minkowski’s inequality.},
author = {Matkowski, J., Świątkowski, T.},
journal = {Fundamenta Mathematicae},
keywords = {subadditive function; homeomorphisms of $ℝ_+$; Mulholland's inequality; convex function; iteration; measure space; the converse of Minkowski's inequality; subadditive; convex homeomorphisms; quasi-addition; Mulholland inequalities; Minkowski inequalities},
language = {eng},
number = {1},
pages = {75-85},
title = {Subadditive functions and partial converses of Minkowski's and Mulholland's inequalities},
url = {http://eudml.org/doc/211993},
volume = {143},
year = {1993},
}

TY - JOUR
AU - Matkowski, J.
AU - Świątkowski, T.
TI - Subadditive functions and partial converses of Minkowski's and Mulholland's inequalities
JO - Fundamenta Mathematicae
PY - 1993
VL - 143
IS - 1
SP - 75
EP - 85
AB - Let ϕ be an arbitrary bijection of $ℝ_+$. We prove that if the two-place function $ϕ^{-1}[ϕ (s)+ϕ (t)]$ is subadditive in $ℝ^2_+$ then $ϕ$ must be a convex homeomorphism of $ℝ_+$. This is a partial converse of Mulholland’s inequality. Some new properties of subadditive bijections of $ℝ_+$ are also given. We apply the above results to obtain several converses of Minkowski’s inequality.
LA - eng
KW - subadditive function; homeomorphisms of $ℝ_+$; Mulholland's inequality; convex function; iteration; measure space; the converse of Minkowski's inequality; subadditive; convex homeomorphisms; quasi-addition; Mulholland inequalities; Minkowski inequalities
UR - http://eudml.org/doc/211993
ER -

## References

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1. [1] J. Aczél, Lectures on Functional Equations and Their Applications, Academic Press, New York 1966. Zbl0139.09301
2. [2] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Polish Scientific Publishers and Silesian University, Warszawa-Kraków-Katowice 1985.
3. [3] J. Matkowski, The converse of the Minkowski's inequality theorem and its generalization, Proc. Amer. Math. Soc. 109 (1990), 663-675. Zbl0704.46020
4. [4] J. Matkowski and T. Świątkowski, Quasi-monotonicity, subadditive bijections of ${ℝ}_{+}$, and characterization of ${L}^{p}$-norm, J. Math. Anal. Appl. 154 (1991), 493-506. Zbl0725.39007
5. [5] J. Matkowski and T. Świątkowski, On subadditive functions, Proc. Amer. Math. Soc., to appear. Zbl0785.39004
6. [6] H. P. Mulholland, On generalizations of Minkowski's inequality in the form of a triangle inequality, Proc. London Math. Soc. 51 (1950), 294-307. Zbl0035.03501

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