On a functional equation related to two-variable Cauchy means

Kiss, Tibor

doi:10.7153/mia-2019-22-76

Cited by 4 publications

(5 citation statements)

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“…The implications (viii) ⇒ (ix) and (ix) ⇒ (i) are obvious. Finally, the equivalence of the assertions (vii) and (ix) is a consequence of the main result of the paper [12].…”

Section: Necessary and Sufficient Conditions For The Equality Of Genementioning

confidence: 67%

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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“…The implications (viii) ⇒ (ix) and (ix) ⇒ (i) are obvious. Finally, the equivalence of the assertions (vii) and (ix) is a consequence of the main result of the paper [12].…”

Section: Necessary and Sufficient Conditions For The Equality Of Genementioning

confidence: 67%

On the equality of two-variable general functional means

Losonczi

Zakaria

2020

Aequat. Math.

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“…The result requiring the weakest regularity conditions known today can be found in [22], where the same solution was obtained as in [4] and [23] but assuming only that ϕ is continuous. To make this result easier to formulate, we introduce the following notation.…”

Section: The Invariance Equation and Its Reformulationmentioning

confidence: 82%

Regular solutions of a functional equation derived from the invariance problem of Matkowski means

Kiss¹

2022

Preprint

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The main result of the present paper is about the solutions of the functional equation Fx + y 2 + f1(x) + f2(y) = G(g1(x) + g2(y)), x, y ∈ I, derived originally, in a natural way, from the invariance problem of generalized weighted quasi-arithmetic means, where F, f1, f2, g1, g2 : I → R and G : g1(I) + g2(I) → R are the unknown functions assumed to be continuously differentiable with 0 / ∈ g ′ 1 (I) ∪ g ′ 2 (I), and the set I stands for a nonempty open subinterval of R. In addition to these, we will also touch upon solutions not necessarily regular. More precisely, we are going to solve the above equation assuming first that F is affine on I and g1 and g2 are continuous functions strictly monotone in the same sense, and secondly that g1 and g2 are invertible affine functions with a common additive part.We shall say that f is locally affine on J or locally constant on J if f is affine or constant on some subinterval U ⊆ J of positive length different from J. If f is neither affine (resp. constant) nor locally affine (resp. locally constant) on J, then it will be called nowhere affine (resp. nowhere constant) on J.We say that a two-place function M : J × J → R is a two-variable mean on J or, shortly, a mean ifElementary examples of means are the harmonic mean and the geometric mean over the set of positive reals, and the arithmetic mean. Rather than giving furter examples, we immediately introduce a general family of twovariable means, which covers the previous examples and which, incidentally, is also the object of our investigation.A mean M : J ×J → R will be called a generalized weighted quasi-arithmetic mean if one can find continuous functions f, g : J → R strictly monotone in the same sense such that (0.1)where the functions f and g are called the generators of the mean in question. If the generators differ on J in an additive constant, then we come to the notion of quasi-arithmetic means. To see that our previous examples are

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“…The equality of Cauchy means to two-variable quasiarithmetic means has been characterized by Kiss and Páles [8].…”

Section: Introductionmentioning

confidence: 99%