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1
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On an elementary inclusion problem and generalized weighted quasi-arithmetic means

100%
Daróczy Z., Páles Z.
Banach Center Publications
|
2013
|
tom 99
|
nr 1
45-54
EN
The aim of this note is to characterize the real coefficients p₁,...,pₙ and q₁,...,qₖ so that $∑_{i=1}^{n} p_ix_i + ∑_{j=1}^{k} q_jy_j ∈ conv{x₁,...,xₙ}$ be valid whenever the vectors x₁,...,xₙ, y₁,...,yₖ satisfy {y₁,...,yₖ} ⊆ conv{x₁,...,xₙ}. Using this characterization, a class of generalized weighted quasi-arithmetic means is introduced and several open problems are formulated.
2
Content available remote

Remarks on the comparison of weighted quasi-arithmetic means

100%
Maksa G., Páles Z.
Colloquium Mathematicum
|
2010
|
tom 120
|
nr 1
77-84
EN
We present comparison theorems for the weighted quasi-arithmetic means and for weighted Bajraktarević means without supposing in advance that the weights are the same.
3
Content available remote

Implications between approximate convexity properties and approximate Hermite-Hadamard inequalities

100%
Makó J., Páles Z.
Open Mathematics
|
2012
|
tom 10
|
nr 3
1017-1041
EN
The connection between the functional inequalities $$f\left( {\frac{{x + y}} {2}} \right) \leqslant \frac{{f\left( x \right) + f\left( y \right)}} {2} + \alpha _J \left( {x - y} \right), x,y \in D,$$ and $$\int_0^1 {f\left( {tx + \left( {1 - t} \right)y} \right)\rho \left( t \right)dt \leqslant \lambda f\left( x \right) + \left( {1 - \lambda } \right)f\left( y \right) + \alpha _{\rm H} \left( {x - y} \right),} x,y \in D,$$ 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.
4
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Minkowski sums of Cantor-type sets

100%
Nikodem K., Páles Z.
Colloquium Mathematicum
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2010
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tom 119
|
nr 1
95-108
EN
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.
5
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Abstract separation theorems of Rodé type and their applications

81%
Nikodem K., Páles Z., Wąsowicz S.
Annales Polonici Mathematici
|
1999
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tom 72
|
nr 3
207-217
EN
Sufficient and necessary conditions are presented under which two given functions can be separated by a function Π-affine in Rodé sense (resp. Π-convex, Π-concave). As special cases several old and new separation theorems are obtained.
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