# Convex-like inequality, homogeneity, subadditivity, and a characterization of ${L}^{p}$-norm

• Volume: 60, Issue: 3, page 221-230
• ISSN: 0066-2216

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## Abstract

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Let a and b be fixed real numbers such that 0 < mina,b < 1 < a + b. We prove that every function f:(0,∞) → ℝ satisfying f(as + bt) ≤ af(s) + bf(t), s,t > 0, and such that $limsu{p}_{t\to 0+}f\left(t\right)\le 0$ must be of the form f(t) = f(1)t, t > 0. This improves an earlier result in [5] where, in particular, f is assumed to be nonnegative. Some generalizations for functions defined on cones in linear spaces are given. We apply these results to give a new characterization of the ${L}^{p}$-norm.

## How to cite

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Janusz Matkowski, and Marek Pycia. "Convex-like inequality, homogeneity, subadditivity, and a characterization of $L^p$-norm." Annales Polonici Mathematici 60.3 (1995): 221-230. <http://eudml.org/doc/262515>.

@article{JanuszMatkowski1995,
abstract = {Let a and b be fixed real numbers such that 0 < mina,b < 1 < a + b. We prove that every function f:(0,∞) → ℝ satisfying f(as + bt) ≤ af(s) + bf(t), s,t > 0, and such that $limsup_\{t → 0+\} f(t) ≤ 0$ must be of the form f(t) = f(1)t, t > 0. This improves an earlier result in [5] where, in particular, f is assumed to be nonnegative. Some generalizations for functions defined on cones in linear spaces are given. We apply these results to give a new characterization of the $L^p$-norm.},
author = {Janusz Matkowski, Marek Pycia},
journal = {Annales Polonici Mathematici},
keywords = {functional inequality; subadditive functions; homogeneous functions; Banach functionals; convex functions; linear space; cones; measure space; integrable step functions; $L^p$-norm; Minkowski's inequality; convexity; subadditivity; upper limit; boundedness; norm; homogeneity; linear functions; convex cones; -norm},
language = {eng},
number = {3},
pages = {221-230},
title = {Convex-like inequality, homogeneity, subadditivity, and a characterization of $L^p$-norm},
url = {http://eudml.org/doc/262515},
volume = {60},
year = {1995},
}

TY - JOUR
AU - Janusz Matkowski
AU - Marek Pycia
TI - Convex-like inequality, homogeneity, subadditivity, and a characterization of $L^p$-norm
JO - Annales Polonici Mathematici
PY - 1995
VL - 60
IS - 3
SP - 221
EP - 230
AB - Let a and b be fixed real numbers such that 0 < mina,b < 1 < a + b. We prove that every function f:(0,∞) → ℝ satisfying f(as + bt) ≤ af(s) + bf(t), s,t > 0, and such that $limsup_{t → 0+} f(t) ≤ 0$ must be of the form f(t) = f(1)t, t > 0. This improves an earlier result in [5] where, in particular, f is assumed to be nonnegative. Some generalizations for functions defined on cones in linear spaces are given. We apply these results to give a new characterization of the $L^p$-norm.
LA - eng
KW - functional inequality; subadditive functions; homogeneous functions; Banach functionals; convex functions; linear space; cones; measure space; integrable step functions; $L^p$-norm; Minkowski's inequality; convexity; subadditivity; upper limit; boundedness; norm; homogeneity; linear functions; convex cones; -norm
UR - http://eudml.org/doc/262515
ER -

## References

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1. [1] J. Aczél and J. Dhombres, Functional Equations in Several Variables, Encyclopedia Math. Appl. 31, Cambridge University Press, Cambridge, Sydney, 1989. Zbl0685.39006
2. [2] E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, Amer. Math. Soc. Colloq. Publ. 31, Amer. Math. Soc., Providence, R.I., 1957. Zbl0078.10004
3. [3] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities. Cauchy's equation and Jensen's inequality, Prace Nauk. Uniw. Śl. 489, Polish Scientific Publishers, 1985.
4. [4] J. Matkowski, On a characterization of ${L}^{p}$-norm, Ann. Polon. Math. 50 (1989), 137-144. Zbl0703.46022
5. [5] J. Matkowski, A functional inequality characterizing convex functions, conjugacy and a generalization of Hölder's and Minkowski's inequalities, Aequationes Math. 40 (1990), 168-180. Zbl0715.39013
6. [6] J. Matkowski, Functional inequality characterizing nonnegative concave functions in (0,∞), ibid. 43 (1992), 219-224. Zbl0756.39017
7. [7] J. Matkowski, The converse of the Minkowski's inequality theorem and its generalization, Proc. Amer. Math. Soc. 109 (1990), 663-675. Zbl0704.46020
8. [8] J. Matkowski, ${L}^{p}$-like paranorms, in: Selected Topics in Functional Equations and Iteration Theory, Proc. Austrian-Polish Seminar, Graz, 1991, Grazer Math. Ber. 316 (1992), 103-135.

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