Wyniki wyszukiwania

1

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

100%

Matkowski J., Świątkowski T.

Fundamenta Mathematicae

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1993

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tom 143

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nr 1

75-85

EN

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.

2

The converse of the Hölder inequality and its generalizations

100%

Matkowski J.

Studia Mathematica

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1994

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tom 109

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nr 2

171-182

EN

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.

3

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

67%

Matkowski J., Pycia M.

Annales Polonici Mathematici

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1994-1995

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tom 60

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nr 3

221-230

EN

Let a and b be fixed real numbers such that 0 < min{a,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.

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Fundamenta Mathematicae

1 Studia Mathematica

3 Matkowski J.

1 Pycia M.

1 Świątkowski T.

1 1995

1 1994

1 1993

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

The converse of the Hölder inequality and its generalizations

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