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On Baire measurable solutions of some functional equations

100%
Baron K.
Open Mathematics
|
2009
|
tom 7
|
nr 4
804-808
EN
We establish conditions under which Baire measurable solutions f of $$ \Gamma (x,y,|f(x) - f(y)|) = \Phi (x,y,f(x + \phi _1 (y)),...,f(x + \phi _N (y))) $$ defined on a metrizable topological group are continuous at zero.
2
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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.
3
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On Popoviciu-Ionescu Functional Equation

76%
Almira J. M.
Annales Mathematicae Silesianae
|
2016
|
tom 30
|
nr 1
5-15
EN
We study a functional equation first proposed by T. Popoviciu [15] in 1955. It was solved for the easiest case by Ionescu [9] in 1956 and, for the general case, by Ghiorcoiasiu and Roscau [7] and Radó [17] in 1962. Our solution is based on a generalization of Radó’s theorem to distributions in a higher dimensional setting and, as far as we know, is different than existing solutions. Finally, we propose several related open problems.
4
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Weighted entropies

64%
Ebanks B.
Open Mathematics
|
2010
|
tom 8
|
nr 3
602-615
EN
We present an axiomatic characterization of entropies with properties of branching, continuity, and weighted additivity. We deliberately do not assume that the entropies are symmetric. The resulting entropies are generalizations of the entropies of degree α, including the Shannon entropy as the case α = 1. Such “weighted” entropies have potential applications to the “utility of gambling” problem.
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