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The converse of the Hölder inequality and its generalizations

Janusz Matkowski (1994)

Studia Mathematica

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...

The Young inequality and the Δ₂-condition

Philippe Laurençot (2002)

Colloquium Mathematicae

If φ: [0,∞) → [0,∞) is a convex function with φ(0) = 0 and conjugate function φ*, the inequality x y ≤ ε φ ( x ) + C ε φ * ( y ) is shown to hold true for every ε ∈ (0,∞) if and only if φ* satisfies the Δ₂-condition.

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