An Arithmetic and Geometric Mean Invariant

Aequationes mathematicae, 2011

Under some conditions on the functions f and g defined in a real interval I the function

Fuzzy Sets and Systems

Let I ⊂ (0, ∞) be an interval that is closed with respect to the multiplication. The operations C f,g : I 2 → I of the form C f,g (x, y) = (f • g) −1 (f (x) • g (y)) , where f, g are bijections of I are considered. Their connections with generalized weighted quasi-geometric means is presented. It is shown that invariance question within the class of this operations leads to means of iterative type and to a problem on a composite functional equation. An application of the invariance identity to determine effectively the limit of the sequence of iterates of some generalized quasi-geometric mean-type mapping, and the form of all continuous functions which are invariant with respect to this mapping are given. The equality of two considered operations is also discussed. C f,g (x, y) = (f • g) −1 (f (x) • g (y)) ,

Applied Mathematics and Computation, 2001

The notions of prickly set, scalar and vectorial mean are de®ned. A noncontinuous generalization of the arithmetic±geometric mean is given, by considering the recursion x n1 F x n , where F : C 3 C is a vectorial mean and C is a closed prickly subset of R m . The convergence of this recursion is proved and it is shown that the limit is contained in the diagonal of C. If F is continuous, it is deduced that the limit of the recursion is a continuous function of the initial value x x 0 . Denoting the limit by F I x it is proved that if F is monotone, then F I it is also monotone (where the monotonicity is considered with respect to the closed cone R m ). Ó (A. Ek art), [email protected] (S.Z. N emeth).

Mathematics, 2020

We prove that whenever the selfmapping (M1,…,Mp):Ip→Ip, (p∈N and Mi-s are p-variable means on the interval I) is invariant with respect to some continuous and strictly monotone mean K:Ip→I then for every nonempty subset S⊆{1,…,p} there exists a uniquely determined mean KS:Ip→I such that the mean-type mapping (N1,…,Np):Ip→Ip is K-invariant, where Ni:=KS for i∈S and Ni:=Mi otherwise. Moreover min(Mi:i∈S)≤KS≤max(Mi:i∈S). Later we use this result to: (1) construct a broad family of K-invariant mean-type mappings, (2) solve functional equations of invariant-type, and (3) characterize Beta-type means.

Bulletin of the Australian Mathematical Society, 1997

In this paper we introduce some new mappings associated with the weighted geometric mean. These are used to derive structural results linking weighted geometric and arithmetic means.

Results in Mathematics, 1990

ISRN Mathematical Analysis, 2011

We extend the classical notions of translativity and homogeneity of means to F-homogeneity, that is, invariance with respect to an operation F : I × I → I. We find the shape of F for the arithmetic weighted mean and then the general form of F for quasi-linear means. Also, we are interested in characterizations of means. It turns out that no quasi-arithmetic mean can be characterized by F-homogeneity with respect to a single operation F, one needs to take two of such operations in order to characterize a mean.

Annales Polonici Mathematici, 2006

Publicationes Mathematicae Debrecen, 2016

Let At, Ht, and Gt denote, respectively, the two-variable weighted arithmetic, harmonic and geometric means with the weight t ∈ (0, 1). Fixing arbitrarily s, t ∈ (0, 1), and choosing for K one of these three means of weight s, and for M another mean of weight t, we examine when the function N satisfying the equality K•(M, N ) = K is a mean, that is when the mean K is (M, N )-invariant. The convergence of the iterates of (M, N ) is considered. The obtained results are applied to find the invariant functions with respect to the suitable mean-type mappings.

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.