Invariant and complementary quasi-arithmetic means

Matkowski, Janusz

doi:10.1007/s000100050072

Cited by 85 publications

(81 citation statements)

References 7 publications

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“…Given a mean-type mapping M : I → I and a mean K : I → I, we say that [6] K is invariant with respect to the mean-type mapping M, briefly M-invariant, if…”

Section: Remark 13mentioning

confidence: 99%

Fixed points and iterations of mean-type mappings

Matkowski

2012

Open Mathematics

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Let (X ) be a metric space and T : X → X a continuous map. If the sequence (T ) ∈N of iterates of T is pointwise convergent in X , then for any ∈ X , the limitis a fixed point of T . The problem of determining the form of µ T leads to the invariance equation µ T • T = µ T , which is difficult to solve in general if the set of fixed points of T is not a singleton. We consider this problem assuming that X = I , where I is a real interval, ≥ 2 a fixed positive integer and T is the mean-type mapping M =(M 1 M ) of I . In this paper we give the explicit forms of µ M for some classes of mean-type mappings. In particular, the classical Pythagorean harmony proportion can be interpreted as an important invariance equality. Some applications are presented. We show that, in general, the mean-type mappings are not non-expansive. MSC:26E60, 26A18

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“…Given a mean-type mapping M : I → I and a mean K : I → I, we say that [6] K is invariant with respect to the mean-type mapping M, briefly M-invariant, if…”

Section: Remark 13mentioning

confidence: 99%

Fixed points and iterations of mean-type mappings

Matkowski

2012

Open Mathematics

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“…In 1998 the functional equation (1.1) was re-discovered by J. Matkowski [8], who formulated the problem more exactly in the following way. 2 …”

Section: History Of the Problemmentioning

confidence: 99%

Extension Theorems for the Matkowski-Suto Problem

Daróczy¹,

Maksa²

2000

Demonstratio Mathematica

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Abstract. In this paper the functional equation is studied, where in a small subinterval of I. History of the ProblemIn 1914 O. Suto published a paper of two parts in the Tohoku Mathematical Journal [11], in which he first examined the functional equationThe author writes the following: "The analytic functions tp,ip and their inverses are supposed to be one-valued, as ever we do;is a mean of x and y in certain sense." In 1998 the functional equation (1.1) was re-discovered by J. Matkowski [8], who formulated the problem more exactly in the following way.1991 Mathematics Subject Classification: 39B22, 39B12, 26A18.

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“…Some conditions guarantying convergence of iterates (K, L) n to a unique (K, L)-invariant mean-type mapping (M, M ) , and M • (K, L) = M ( [7], also [4,6,11]), generalize in particular, the well-known theorem of Gauss [2] on the arithmetic-geometric iterations. If the invariance equality M • (K, L) = M is satisfies one says that the means K and L are mutually M -complementary with respect to M (briefly, M -complementary) [5].…”

Section: Introductionmentioning

confidence: 99%

Explicit Solutions of the Invariance Equation for Means

2016

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Abstract. Extending the notion of projective means we first generalize an invariance identity related to the Carlson log given in Kahlig and Matkowski (Math Inequal Appl 18(3):1143-1150, 2015, and then, more generally, given a bivariate symmetric, homogeneous and monotone mean M , we give explicit formula for a rich family of pairs of M -complementary means. We prove that this method cannot be extended for higher dimension. Some examples are given and two open questions are proposed.Mathematics Subject Classification. 26D15.

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Invariant and complementary quasi-arithmetic means

Cited by 85 publications

References 7 publications

Fixed points and iterations of mean-type mappings

Fixed points and iterations of mean-type mappings

Extension Theorems for the Matkowski-Suto Problem

Explicit Solutions of the Invariance Equation for Means

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