Set-Valued Means

Acta Scientiarum Mathematicarum, 2015

In this paper we characterize generalized quasi-arithmetic means, that is means of the form M (x 1 ,. .. , x n) := (f 1 + • • • + f n) −1 (f 1 (x 1) + • • • + f n (x n)), where f 1 ,. .. , f n : I → R are strictly increasing and continuous functions. Our characterization involves the Gauss composition of the cyclic mean-type mapping induced by M and a generalized bisymmetry equation.

Journal of Mathematical Analysis and Applications, 1997

Ž . Let , , ␥ , ␤ : 0, ϱ ª R, strictly monotonic and continuous functions, be the generators of the positively homogeneous quasi-arithmetic means M , M , M , ␥ and M . The main result gives full characterizations of the functions , , ␥ , and ␤

Results in Mathematics

This paper offers a solution of the functional equation $$\begin{aligned}&\big (tf(x)+(1-t)f(y)\big )\varphi (tx+(1-t)y)\\&\quad =tf(x)\varphi (x)+(1-t)f(y)\varphi (y) \qquad (x,y\in I), \end{aligned}$$(tf(x)+(1-t)f(y))φ(tx+(1-t)y)=tf(x)φ(x)+(1-t)f(y)φ(y)(x,y∈I),where $$t\in \,]0,1[\,$$t∈]0,1[, $$\varphi :I\rightarrow \mathbb {R}$$φ:I→R is strictly monotone, and $$f:I\rightarrow \mathbb {R}$$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.

2000

In this article, the author introduces the generalized abstracted mean values which extend the concepts of most means with two variables, and researches their basic properties and monotonicities.

Publicationes Mathematicae Debrecen

Under some regularity assumptions imposed on the generators f , g, we determine all the quasi-arithmetic means M [f ] , M [g] and all real numbers λ and µ such that λM [f ] + µM [g] = A, where A is the arithmetic mean.

Acta Mathematica Hungarica, 2021

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 .

Information Sciences, 2018

In this paper we analyze the notion of a finite mean from an axiomatic point of view. We discuss several axiomatic alternatives, with the aim of establishing a universal definition reconciling all of them and exploring theoretical links to some branches of Mathematics as well as to multidisciplinary applications.

Colloquium Mathematicum, 2008

In this paper we study the problem of extending means to means of higher order. We show how higher order means can be inductively defined and established in general metric spaces, in particular, in convex metric spaces. As a particular application, we consider the positive operators on a Hilbert space under the Thompson metric and show that the operator logarithmic mean admits extensions of all higher orders, thus providing a positive solution to a problem of Petz and Temesi .

2003

Let M : (0, ∞) 2 → (0, ∞) be a homogeneous strict mean such that the function h := M (·, 1) is twice dierentiable and 0 = h (1) = 1. It is shown that if there exists an M -ane function, continuous at a point which is neither constant nor linear, then M must be a weighted power mean. Moreover the homogeneity condition of M can be replaced by M -convexity of two suitably chosen linear functions. With the aid of iteration groups, some generalizations characterizing the weighted quasi-arithmetic means are given. A geometrical aspect of these results is discussed.

Colloquium Mathematicum, 2013

A generalization of the weighted quasi-arithmetic mean generated by continuous and increasing (decreasing) functions f1,. .. , f k : I → R, k ≥ 2, denoted by A [f 1 ,...,f k ] , is considered. Some properties of A [f 1 ,...,f k ] , including "associativity" assumed in the Kolmogorov-Nagumo theorem, are shown. Convex and affine functions involving this type of means are considered. Invariance of a quasi-arithmetic mean with respect to a special mean-type mapping built of generalized means is applied in solving a functional equation. For a sequence of continuous strictly increasing functions fj : I → R, j ∈ N, a mean A [f 1 ,f 2 ,...] : ∞ k=1 I k → I is introduced and it is observed that, except symmetry, it satisfies all conditions of the Kolmogorov-Nagumo theorem. A problem concerning a generalization of this result is formulated.