On a functional equation related to two-variable weighted quasi-arithmetic means

Kiss, Tibor

doi:10.1080/10236198.2017.1397140

Cited by 12 publications

(16 citation statements)

References 44 publications

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“…Substituting y := x into the last equality, (9) follows immediately. Differentiating (8) with respect to x, we obtain…”

Section: Solution Of the Fundamental Functional Equation (3)mentioning

confidence: 99%

“…where f, ϕ : I → R and t ∈ ]0, 1[ is fixed. This equation was considered and solved in the case t = 1 2 in [5] under strict monotonicity and continuity of ϕ and in [8] under continuity of ϕ, respectively. In Theorem 4 and Theorem 5 below, we completely solve (3) assuming only the strict monotonicity of ϕ and also including the case t = 1 2 .…”

Section: Introductionmentioning

confidence: 99%

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On the Equality of Bajraktarević Means to Quasi-Arithmetic Means

Zakaria

2019

Results Math

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“…Substituting y := x into the last equality, (9) follows immediately. Differentiating (8) with respect to x, we obtain…”

Section: Solution Of the Fundamental Functional Equation (3)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the Equality of Bajraktarević Means to Quasi-Arithmetic Means

Zakaria

2019

Results Math

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“…Quasi-arithmetic means are central objects in theory of functional equations, in particular in the theory of means (see e.g. [6], [10], [12], [13], [14], [17], [21], [22], [23], [24], [25]).…”

Section: Introductionmentioning

confidence: 99%

Characterization of quasi-arithmetic means without regularity condition

2021

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In this paper we show that bisymmetry, which is an algebraic property, has a regularity improving feature. More precisely, we prove that every bisymmetric, partially strictly monotonic, reflexive and symmetric function F : I 2 → I is continuous. As a consequence, we obtain a finer characterization of quasiarithmetic means than the classical results of Aczél [1], Kolmogoroff [18], Nagumo [20] and de Finetti [11].

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“…A similar problem, the mixed equality problem of quasiarithmetic and Lagrangian means was solved by Páles [24]. Another mixed equality problem, the equality of twovariable quasiarithmetic and Bajraktarevic means in the symmetric and in the weighted setting was also solved by Daróczy-Maksa-Páles [8] and by Kiss-Páles [11], respectively.…”

Section: Introductionmentioning

confidence: 99%

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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On a functional equation related to two-variable weighted quasi-arithmetic means

Cited by 12 publications

References 44 publications

On the Equality of Bajraktarević Means to Quasi-Arithmetic Means

On the Equality of Bajraktarević Means to Quasi-Arithmetic Means

Characterization of quasi-arithmetic means without regularity condition

On the equality of two-variable general functional means

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