On the Equality of Bajraktarević Means to Quasi-Arithmetic Means

Zakaria, Amr

doi:10.1007/s00025-019-1141-5

Cited by 11 publications

(9 citation statements)

References 10 publications

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“…The existence of some γ such that the identity W 1,0 F,G = γW 1,0 f,g holds is a direct consequence of the integration of the equality Φ f,g = Φ F,G . Applying implication (iv) ⇒ (ii) of [26,Theorem 10], we conclude that there exist real constants a, b, c, A, B, C such that the equalities (36) hold. Therefore, assertion (v) is valid.…”

Section: Necessary and Sufficient Conditions For The Equality Of Genementioning

confidence: 87%

“…The implications (viii) ⇒ (ix) and (ix) ⇒ (i) are obvious. Finally, the equivalence of (vii) and (ix) is a consequence of [26,Corollary 9]. Proof.…”

Section: Necessary and Sufficient Conditions For The Equality Of Genementioning

confidence: 90%

“…To prove the implication (v) ⇒ (viii), assume that (v) holds for some constants a, b, c, A, B, C, γ ∈ R. The implication (ii) ⇒ (iii) of [26,Theorem 10] implies that M f,g;μ = A ϕ and M F,G;μ = A ψ hold on I 2 with ϕ = W 1,0 f,g and ψ = W 1,0 F,G . Thus, using that W 1,0 f,g = γW 1,0 F,G , we get ϕ = γψ.…”

Section: Necessary and Sufficient Conditions For The Equality Of Genementioning

confidence: 99%

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On the equality of two-variable general functional means

Losonczi

Zakaria

2020

Aequat. Math.

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Given two functions $$f,g:I\rightarrow \mathbb {R}$$ f , g : I → R and a probability measure $$\mu $$ μ on the Borel subsets of [0, 1], the two-variable mean $$M_{f,g;\mu }:I^2\rightarrow I$$ M f , g ; μ : I 2 → I is defined by $$\begin{aligned} M_{f,g;\mu }(x,y) :=\bigg (\frac{f}{g}\bigg )^{-1}\left( \frac{\int _0^1 f\big (tx+(1-t)y\big )d\mu (t)}{\int _0^1 g\big (tx+(1-t)y\big )d\mu (t)}\right) \quad (x,y\in I). \end{aligned}$$ M f , g ; μ ( x , y ) : = ( f g ) - 1 ∫ 0 1 f ( t x + ( 1 - t ) y ) d μ ( t ) ∫ 0 1 g ( t x + ( 1 - t ) y ) d μ ( t ) ( x , y ∈ I ) . This class of means includes quasiarithmetic as well as Cauchy and Bajraktarević means. The aim of this paper is, for a fixed probability measure $$\mu $$ μ , to study their equality problem, i.e., to characterize those pairs of functions (f, g) and (F, G) for which $$\begin{aligned} M_{f,g;\mu }(x,y)=M_{F,G;\mu }(x,y) \quad (x,y\in I) \end{aligned}$$ M f , g ; μ ( x , y ) = M F , G ; μ ( x , y ) ( x , y ∈ I ) holds. Under at most sixth-order differentiability assumptions for the unknown functions f, g and F, G, we obtain several necessary conditions in terms of ordinary differential equations for the solutions of the above equation. For two particular measures, a complete description is obtained. These latter results offer eight equivalent conditions for the equality of Bajraktarević means and of Cauchy means.

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Section: Necessary and Sufficient Conditions For The Equality Of Genementioning

confidence: 87%

“…The implications (viii) ⇒ (ix) and (ix) ⇒ (i) are obvious. Finally, the equivalence of (vii) and (ix) is a consequence of [26,Corollary 9]. Proof.…”

Section: Necessary and Sufficient Conditions For The Equality Of Genementioning

confidence: 90%

Section: Necessary and Sufficient Conditions For The Equality Of Genementioning

confidence: 99%

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On the equality of two-variable general functional means

Losonczi

Zakaria

2020

Aequat. Math.

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“…is an example of a homogeneous mean (i.e., L(λx, λy) = λL(x, y) for any λ > 0) that is not a QAM. Besides the family of QAMs, there exist many other families of means [5]: For example, let us mention the Lagrangean means [18] which intersect with the QAMs only for the arithmetic mean, or a generalization of the QAMs called the the Bajraktarević means [35].…”

Section: The Quasi-arithmetic α-Divergencesmentioning

confidence: 99%

The α-divergences associated with a pair of strictly comparable quasi-arithmetic means

Nielsen

2020

Preprint

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We generalize the family of α-divergences using a pair of strictly comparable weighted means. In particular, we obtain the 1-divergence in the limit case α → 1 (a generalization of the Kullback-Leibler divergence) and the 0-divergence in the limit case α → 0 (a generalization of the reverse Kullback-Leibler divergence). We state the condition for a pair of quasi-arithmetic means to be strictly comparable, and report the formula for the quasi-arithmetic α-divergences and its subfamily of bipower homogeneous α-divergences which belong to the Csisár's f -divergences. Finally, we show that these generalized quasi-arithmetic 1-divergences and 0-divergences can be decomposed as the sum of generalized cross-entropies minus entropies, and rewritten as conformal Bregman divergences using monotone embeddings.

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“…For the equality of two Bajraktarević means with more than two variables the equivalence of their generators is not only sufficient but it is also necessary (cf. [1][2][3]14,15]). Surprisingly, in the setting of two-variable means, the situation is completely different.…”

Section: Introductionmentioning

confidence: 99%

On the equality problem of two-variable Bajraktarević means under first-order differentiability assumptions

Zakaria

2022

Aequat. Math.

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The equality problem of two-variable Bajraktarević means can be expressed as the functional equation $$\begin{aligned} \left( \frac{f}{g}\right) ^{-1}\bigg (\frac{f(x)+f(y)}{g(x)+g(y)}\bigg ) =\left( \frac{h}{k}\right) ^{-1}\bigg (\frac{h(x)+h(y)}{k(x)+k(y)}\bigg )\qquad (x,y\in I), \end{aligned}$$ f g - 1 ( f ( x ) + f ( y ) g ( x ) + g ( y ) ) = h k - 1 ( h ( x ) + h ( y ) k ( x ) + k ( y ) ) ( x , y ∈ I ) , where I is a nonempty open real interval, $$f,g,h,k:I\rightarrow {\mathbb {R}}$$ f , g , h , k : I → R are continuous functions, g, k are positive and f/g, h/k are strictly monotone. This functional equation, for the first time, was solved by Losonczi in 1999 under 6th-order continuous differentiability assumptions. Additional and new characterizations of this equality problem have been found recently by Losonczi, Páles and Zakaria under the same regularity assumptions in 2021. In this paper it is shown that the same conclusion can be obtained under substantially weaker regularity conditions, namely, assuming only first-order differentiability.

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On the Equality of Bajraktarević Means to Quasi-Arithmetic Means

Abstract: This paper offers a solution of the functional equationwhere t ∈ ]0, 1[ , ϕ : I → R is strictly monotone, and f : I → R is an arbitrary unknown function. As an immediate application, we shed new light on the equality problem of Bajraktarević means with quasi-arithmetic means.

Cited by 11 publications

References 10 publications

On the equality of two-variable general functional means

On the equality of two-variable general functional means

The α-divergences associated with a pair of strictly comparable quasi-arithmetic means

On the equality problem of two-variable Bajraktarević means under first-order differentiability assumptions

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