On an equation involving weighted quasi-arithmetic means

Let I ? ? be an interval and κ, λ ∈ ? / {0, 1}, µ, ν ∈ (0, 1). We find all pairs (φ, ψ) of continuous and strictly monotonic functions mapping I into ? and satisfying the functional equation $$ \kappa x + (1 - \kappa )y = \lambda \phi ^{ - 1} (\mu \phi (x) + (1 - \mu )\phi (y)) + (1 - \lambda )\psi ^{ - 1} (\nu \psi (x) + (1 - \nu )\psi (y)) $$ which generalizes the Matkowski-Sutô equation. The paper completes a research stemming in the theory of invariant means.     

A functional equation involving comparable weighted quasi-arithmetic means

We discuss the functional equation $$f(M(x,y))=f(N(x,y)) \qquad (x,y\in I) $$ where I is a nonvoid open subinterval of the set of real numbers ?, f:?I??? is an unknown function, and M, N are weighted quasi-arithmetic means on?I. Our typical assumption will be that M and N are comparable. Complete solution will be presented under the additional supposition that f is the product of the generating functions of?M and?N, respectively.     

The Matkowski–Sutő problem for weighted quasi-arithmetic means

In this paper the so-called invariance equation is studied for weighted nonsymmetric quasi-arithmetic means and solved under continuous differentiability assumptions with respect to the generating functions of the quasi-arithmetic means. This revised version was published online in June 2006 with corrections to the Cover Date.     

On a family of quasi-arithmetic means

Summary The main goal of this paper is to solve the idempotency equationF(x, x) = x, x [0, 1] for a class of functions of the type convex linear combination of at-norm and at-conorm. In the non-strict Archimedean case and for eachk (0, 1), we obtain a unique solutionF _k for this equation. While these functionsF _k are not associative, we also prove that they satisfy the bisymmetry condition.     

Characterization of quasi-arithmetic means without regularity condition

Acta Mathematica Hungarica - We show that bisymmetry, which is an algebraic property, has a regularity improving feature. More precisely, we prove that every bisymmetric, partially strictly...     

Invariance of quasi-arithmetic means with function weights

Let I⊂R be a non-trivial interval and let . We present some results concerning the following functional equation, generalizing the Matkowski-Sutô equation,

λ(x,y)φ⁻¹(μ(x,y)φ(x)+(1−μ(x,y))φ(y))+(1−λ(x,y))ψ⁻¹(ν(x,y)ψ(x)+(1−ν(x,y))ψ(y))=λ(x,y)x+(1−λ(x,y))y,     

On the equality of quasi-arithmetic and Lagrangian means

The paper deals with the equality problem of quasi-arithmetic and Lagrangian means which is to determine all pairs of continuous strictly monotone functions φ,ψ:I→R such that, for all x,y∈I,

     

Limit properties in a family of quasi-arithmetic means

It is known that the Power Means tend to the maximum of their arguments when the exponents tend to \({+\infty}\). We give certain necessary and sufficient conditions for a 1-parameter family of quasi-arithmetic means generated by functions satisfying certain smoothness conditions to have an analogous property. Our results are deeply connected with operators introduced by Mikusiński and Páles in the late 1940s and late 1980s, respectively. The main result is a generalization of the author’s earlier results.     

Lower estimation of the difference between quasi-arithmetic means

In the 1960s Cargo and Shisha introduced a metric in a family of quasi-arithmetic means defined on a common interval as the maximal possible difference between these means taken over all admissible vectors with corresponding weights. During the years 2013–2016 we proved that, having two quasi-arithmetic means, we can majorize the distance between them in terms of the Arrow–Pratt index. In this paper we are going to prove that this operator can also be used to establish certain lower bounds of this distance.     

A variant of Jensen-Steffensen's inequality and quasi-arithmetic means

A variant of Jensen-Steffensen's inequality is proved. Necessary and sufficient conditions for the equality in Jensen-Steffensen's inequality are established. Several inequalities involving more than two monotonic functions and generalized quasi-arithmetic means with not only positive weights are proved. It is shown that such generalized quasi-arithmetic means have the same comparability properties as those with positive weights.     

On the characterization of distributivity equations about quasi-arithmetic means

The aim of this paper is to characterize the distributivity equations between quasi-arithmetic means and some binary operators, such as, nullnorms, semi-nullnorms and semi-t-operators. It is shown that the distributivity equations between nullnorms (semi-nullnorms) and quasi-arithmetic means degenerate into the distributivity equations between t-norms or t-conorms (semi-t-norms or semi-t-conorms) and quasi-arithmetic means. But for semi-t-operators, the distributive equations have new solutions. These new results extend the previous ones about the distributivity between t-norms (t-conorms) and quasi-arithmetic means.     

An invariance of geometric mean with respect to generalized quasi-arithmetic means

In this paper, we study the invariance of the geometric mean with respect to some generalized quasi-arithmetic means, namely, we present some results concerning the functional equation

     

The invariance of the arithmetic mean with respect to generalized quasi-arithmetic means

The aim of this paper is to find those pairs of generalized quasi-arithmetic means on an open real interval I for which the arithmetic mean is invariant, i.e., to characterize those continuous strictly monotone functions φ,ψ:I→R and Borel probability measures μ,ν on the interval [0,1] such that