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Implications between approximate convexity properties and approximate Hermite-Hadamard inequalities

Judit MakóZsolt Páles — 2012

Open Mathematics

The connection between the functional inequalities f x + y 2 f x + f y 2 + α J x - y , x , y D , and 0 1 f t x + 1 - t y ρ t d t λ f x + 1 - λ f y + α H x - y , x , y D , is investigated, where D is a convex subset of a linear space, f: D → ℝ, α H;α J: D-D → ℝ are even functions, λ ∈ [0; 1], and ρ: [0; 1] →ℝ+ is an integrable nonnegative function with ∫01 ρ(t) dt = 1.

Minkowski sums of Cantor-type sets

Kazimierz NikodemZsolt Páles — 2010

Colloquium Mathematicae

The classical Steinhaus theorem on the Minkowski sum of the Cantor set is generalized to a large class of fractals determined by Hutchinson-type operators. Numerous examples illustrating the results obtained and an application to t-convex functions are presented.

On an elementary inclusion problem and generalized weighted quasi-arithmetic means

Zoltán DaróczyZsolt Páles — 2013

Banach Center Publications

The aim of this note is to characterize the real coefficients p₁,...,pₙ and q₁,...,qₖ so that i = 1 n p i x i + j = 1 k q j y j c o n v x , . . . , x be valid whenever the vectors x₁,...,xₙ, y₁,...,yₖ satisfy y₁,...,yₖ ⊆ convx₁,...,xₙ. Using this characterization, a class of generalized weighted quasi-arithmetic means is introduced and several open problems are formulated.

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