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On subadditive functions and ψ-additive mappings

100%

Matkowski J.

Open Mathematics

2003

tom 1

nr 4

435-440

In [4], assuming among others subadditivity and submultiplicavity of a function ψ: [0, ∞)→[0, ∞), the authors proved a Hyers-Ulam type stability theorem for “ψ-additive” mappings of a normed space into a normed space. In this note we show that the assumed conditions of the function ψ imply that ψ=0 and, consequently, every “ψ-additive” mapping must be additive

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

86%

Matkowski J., Świątkowski T.

Fundamenta Mathematicae

1993

tom 143

nr 1

75-85

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.

The converse of the Hölder inequality and its generalizations

86%

Matkowski J.

Studia Mathematica

1994

tom 109

nr 2

171-182

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.

Ograniczanie wyników

1 Fundamenta Mathematicae

1 Open Mathematics

1 Studia Mathematica

3 Matkowski J.

1 Świątkowski T.

1 2003

1 1994

1 1993

Wyniki wyszukiwania

On subadditive functions and ψ-additive mappings

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

The converse of the Hölder inequality and its generalizations