The converse of the Hölder inequality and its generalizations

Janusz Matkowski

Studia Mathematica (1994)

  • Volume: 109, Issue: 2, page 171-182
  • ISSN: 0039-3223

Abstract

top
Let (Ω,Σ,μ) be a measure space with two sets A,B ∈ Σ such that 0 < μ (A) < 1 < μ (B) < ∞ and suppose that ϕ and ψ are arbitrary bijections of [0,∞) such that ϕ(0) = ψ(0) = 0. The main result says that if ʃ Ω x y d μ ϕ - 1 ( ʃ Ω ϕ x d μ ) ψ - 1 ( ʃ Ω ψ x d μ ) for all μ-integrable nonnegative step functions x,y then ϕ and ψ must be conjugate power functions. If the measure space (Ω,Σ,μ) has one of the following properties: (a) μ (A) ≤ 1 for every A ∈ Σ of finite measure; (b) μ (A) ≥ 1 for every A ∈ Σ of positive measure, then there exist some broad classes of nonpower bijections ϕ and ψ such that the above inequality holds true. A general inequality which contains integral Hölder and Minkowski inequalities as very special cases is also given.

How to cite

top

Matkowski, Janusz. "The converse of the Hölder inequality and its generalizations." Studia Mathematica 109.2 (1994): 171-182. <http://eudml.org/doc/216067>.

@article{Matkowski1994,
abstract = {Let (Ω,Σ,μ) be a measure space with two sets A,B ∈ Σ such that 0 < μ (A) < 1 < μ (B) < ∞ and suppose that ϕ and ψ are arbitrary bijections of [0,∞) such that ϕ(0) = ψ(0) = 0. The main result says that if $ʃ_Ω xydμ ≤ ϕ^\{-1\} (ʃ_\{Ω\} ϕ∘x dμ) ψ^\{-1\} (ʃ_\{Ω\} ψ∘x dμ)$ for all μ-integrable nonnegative step functions x,y then ϕ and ψ must be conjugate power functions. If the measure space (Ω,Σ,μ) has one of the following properties: (a) μ (A) ≤ 1 for every A ∈ Σ of finite measure; (b) μ (A) ≥ 1 for every A ∈ Σ of positive measure, then there exist some broad classes of nonpower bijections ϕ and ψ such that the above inequality holds true. A general inequality which contains integral Hölder and Minkowski inequalities as very special cases is also given.},
author = {Matkowski, Janusz},
journal = {Studia Mathematica},
keywords = {measure space; integrable step functions; conjugate functions; a converse of Hölder inequality; subadditive function; convex function; generalized Hölder-Minkowski inequality; integral Hölder and Minkowski inequalities},
language = {eng},
number = {2},
pages = {171-182},
title = {The converse of the Hölder inequality and its generalizations},
url = {http://eudml.org/doc/216067},
volume = {109},
year = {1994},
}

TY - JOUR
AU - Matkowski, Janusz
TI - The converse of the Hölder inequality and its generalizations
JO - Studia Mathematica
PY - 1994
VL - 109
IS - 2
SP - 171
EP - 182
AB - Let (Ω,Σ,μ) be a measure space with two sets A,B ∈ Σ such that 0 < μ (A) < 1 < μ (B) < ∞ and suppose that ϕ and ψ are arbitrary bijections of [0,∞) such that ϕ(0) = ψ(0) = 0. The main result says that if $ʃ_Ω xydμ ≤ ϕ^{-1} (ʃ_{Ω} ϕ∘x dμ) ψ^{-1} (ʃ_{Ω} ψ∘x dμ)$ for all μ-integrable nonnegative step functions x,y then ϕ and ψ must be conjugate power functions. If the measure space (Ω,Σ,μ) has one of the following properties: (a) μ (A) ≤ 1 for every A ∈ Σ of finite measure; (b) μ (A) ≥ 1 for every A ∈ Σ of positive measure, then there exist some broad classes of nonpower bijections ϕ and ψ such that the above inequality holds true. A general inequality which contains integral Hölder and Minkowski inequalities as very special cases is also given.
LA - eng
KW - measure space; integrable step functions; conjugate functions; a converse of Hölder inequality; subadditive function; convex function; generalized Hölder-Minkowski inequality; integral Hölder and Minkowski inequalities
UR - http://eudml.org/doc/216067
ER -

References

top
  1. [1] R. Cooper, Note on the Cauchy-Hölder inequality, J. London Math. Soc. (2) 26 (1928), 8-9. Zbl54.0221.01
  2. [2] Z. Daróczy and Z. Páles, Convexity with given infinite weight sequences, Stochastica 11 (1987), 5-12. 
  3. [3] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge Univ. Press, 1952. 
  4. [4] 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, Warszawa-Kraków-Katowice, 1985. 
  5. [5] N. Kuhn, A note on t-convex functions, in: General Inequalities 4, Internat. Ser. Numer. Math. 71, Birkhäuser, Basel, 1984, 269-276. 
  6. [6] J. Matkowski, Cauchy functional equation on a restricted domain and commuting functions, in: Iteration Theory and Its Functional Equations (Proc. Schloss Hofen, 1984), Lecture Notes in Math. 1163, Springer, Berlin, 1985, 101-106. 
  7. [7] J. Matkowski, Functional inequality characterizing convex functions, conjugacy and a generalization of Hölder's and Minkowski's inequalities, Aequationes Math. 40 (1990), 168-180. Zbl0715.39013
  8. [8] J. Matkowski, The converse of the Minkowski's inequality theorem and its generalization, Proc. Amer. Math. Soc. 109 (1990), 663-675. Zbl0704.46020
  9. [9] J. Matkowski, A generalization of Holder's and Minkowski's inequalities and conjugate functions, in: Constantin Carathéodory: An International Tribute, Vol. II, World Scientific, Singapore, 1991, 819-827. 
  10. [10] J. Matkowski, Functional inequality characterizing concave functions in ( 0 , ) k , Aequationes Math. 43 (1992), 219-224. Zbl0756.39017

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.