Invariance of means
type: Item
instance of: scholarly article
title: Invariance of means
zbMATH Open document ID: 1435.26037
author: Justyna Jarczyk
author: Witold Jarczyk
published in: Aequationes Mathematicae
publication date: 2018-09-26T00:00:00Z
review text: Let \(M\) and \(N\) be means on the same interval \(I\). The paper deals with the following invariance problem: finding a mean \(K\) on \(I\) such that \[ K(M(x,y),N(x,y))=K(x,y), \] for all \(x,y\in I\). One can see as a starting point of this problem the identity \[ \frac{x+y}{2}\cdot \frac{2}{\frac{1}{x}+\frac{1}{y}}=xy. \] In particular, the authors focus their attention on quasi-arithmetic means, Bajraktarević means and Cauchy means. The paper provides an interesting survey of the research on this topic and a comprehensive list of references.
reviewed by: F-C. Mitroi
Mathematics Subject Classification ID: 26E60
Mathematics Subject Classification ID: 39B22
Mathematics Subject Classification ID: 39B12
zbMATH DE Number: 6944063
zbMATH Keywords: Cauchy mean
zbMATH Keywords: Bajraktarević mean
zbMATH Keywords: quasi-arithmetic mean
zbMATH Keywords: Gauss composition
zbMATH Keywords: invariance
MaRDI profile type: Publication
full work available at URL: s00010-018-0564-5
OpenAlex ID: W2861697297
cites work: Q5796678
cites work: Q5514010
cites work: Q5525526
cites work: Q3992445
cites work: Q5789254
cites work: Invariance equation for generalized quasi-arithmetic means
cites work: Computer aided solution of the invariance equation for two-variable Gini means
cites work: Computer aided solution of the invariance equation for two-variable Stolarsky means
cites work: Solving invariance equations involving homogeneous means with the help of computer
cites work: Q3252924
cites work: Q5329230
cites work: Q5577732
cites work: Q3514340
cites work: Iteration of random-valued functions on the unit interval
cites work: A Class of Mean Value Functions
cites work: Lagrangian means
cites work: Interpolating the arithmetic--geometric mean inequality and its operator version
cites work: On the invariance equation for Heinz means
cites work: An invariance of the geometric mean with respect to Stolarsky mean-type mappings
cites work: Q3751634
cites work: Q4441514
cites work: Q3855396
cites work: Q3436133
cites work: Matkowski-Sutô type equation on symmetrized weighted quasi-arithmetic means
cites work: Q5395592
cites work: Q4684609
cites work: Algorithms Involving Arithmetic and Geometric Means
cites work: Q3242256
cites work: Q3622895
cites work: Invariance in the Family of Weighted Gini Means
cites work: Q3836598
cites work: Matkowski-Sutô type problem for conjugate arithmetic means.
cites work: Mean values and functional equations
cites work: On the equality and comparison problem of a class of mean values
cites work: Q3653789
cites work: On the equality problem of conjugate means
cites work: On linear combinations of weighted quasi-arithmetic means
cites work: An extension theorem for a Matkowski–Sutô problem
cites work: Q5634189
cites work: Functional equations on convex sets
cites work: On a problem of Matkowski
cites work: Q4521161
cites work: On two-variable means with variable weights
cites work: On means that are both quasi-arithmetic and conjugate arithmetic
cites work: On a class of means of several variables
cites work: Q4530593
cites work: Q4793074
cites work: The Matkowski-Sutô problem for weighted quasi-arithmetic means
cites work: Q3428703
cites work: Q4432148
cites work: Q5696174
cites work: Q3653882
cites work: A stochastic functional equation
cites work: Invariance of the arithmetic mean with respect to special mean-type mappings
cites work: Q4478690
cites work: The Arithmetic-Harmonic Mean
cites work: Di una formula comprensiva delle medie
cites work: Q5421138

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cites work: A solution of an open problem concerning Lagrangian mean-type mappings

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cites work: Some Cauchy mean-type mappings for which the geometric mean is invariant

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cites work: A square iterative roots of some mean-type mappings

Invariance of means (cites work)
cites work: Q4807503

Invariance of means (cites work)
cites work: An invariance of geometric mean with respect to Lagrangian means

Invariance of means (cites work)
cites work: Q4530594

Invariance of means (cites work)
cites work: Q4758141

Invariance of means (cites work)
cites work: Beiträge zur Störungstheorie der Spektralzerlegung

Invariance of means (cites work)
cites work: Q5786806

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cites work: Regularity properties of functional equations in several variables.

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cites work: Q3531296

Invariance of means (cites work)
cites work: Invariance of quasi-arithmetic means with function weights

Invariance of means (cites work)
cites work: Q3653888

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cites work: On an equation involving weighted quasi-arithmetic means

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cites work: Invariance in a class of Bajraktarević means

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cites work: Determination of conjugate means by reducing to the generalized Matkowski-Sütő equation

Invariance of means (cites work)
cites work: Parametrized means and limit properties of their Gaussian iterations

Invariance of means (cites work)
cites work: Invariance in the class of weighted quasi-arithmetic means

Invariance of means (cites work)
cites work: Q5791292

Invariance of means (cites work)
cites work: Über gewogene Mittelwerte, insbesondere gewogene geometrische Mittel

Invariance of means (cites work)
cites work: Q4480918

Invariance of means (cites work)
cites work: On the composition of homogeneous quasi-arithmetic means

Invariance of means (cites work)
cites work: Invariant means related to classical weighted means

Invariance of means (cites work)
cites work: Sur une équivalence pour les fonctions

Invariance of means (cites work)
cites work: Limit properties of quasi-arithmetic means

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cites work: Q3218572

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cites work: On the compounding of certain means

Invariance of means (cites work)
cites work: Q5637148

Invariance of means (cites work)
cites work: Equality of two variable weighted means: Reduction to differential equations

Invariance of means (cites work)
cites work: Equality of two variable Cauchy mean values

Invariance of means (cites work)
cites work: Q3539371

Invariance of means (cites work)
cites work: The invariance of the arithmetic mean with respect to generalized quasi-arithmetic means

Invariance of means (cites work)
cites work: Remarks on the comparison of weighted quasi-arithmetic means

Invariance of means (cites work)
cites work: On an axiomatization of the quasi-arithmetic mean values without the symmetry axiom

Invariance of means (cites work)
cites work: Invariant and complementary quasi-arithmetic means

Invariance of means (cites work)
cites work: Mean value property and associated functional equations

Invariance of means (cites work)
cites work: Q4947993

Invariance of means (cites work)
cites work: On iteration semigroups of mean-type mappings and invariant means

Invariance of means (cites work)
cites work: Q4530596

Invariance of means (cites work)
cites work: Q3023589

Invariance of means (cites work)
cites work: Lagrangean mean-type mappings for which the arithmetic mean is invariant

Invariance of means (cites work)
cites work: Q5439426

Invariance of means (cites work)
cites work: Q3007316

Invariance of means (cites work)
cites work: Generalized weighted quasi-arithmetic means

Invariance of means (cites work)
cites work: Q2997779

Invariance of means (cites work)
cites work: Quotient mean, its invariance with respect to a quasi-arithmetic mean-type mapping, and some applications

Invariance of means (cites work)
cites work: Invariance of the Cauchy means with respect to Lagrange mean-type mappings of the same generators

Invariance of means (cites work)
cites work: Invariance of a quasi-arithmetic mean with respect to a system of generalized Bajraktarević means

Invariance of means (cites work)
cites work: Invariance of Bajraktarevič mean with respect to quasi arithmetic means

Invariance of means (cites work)
cites work: Halfconvex functions and equality of Daróczy-Páles conjugate means

Invariance of means (cites work)
cites work: Q5395624

Invariance of means (cites work)
cites work: Generalized weighted quasi-arithmetic means and the Kolmogorov-Nagumo theorem

Invariance of means (cites work)
cites work: Invariance of the Bajraktarević means with respect to the Beckenbach-Gini means

Invariance of means (cites work)
cites work: Invariance identity in the class of generalized quasiarithmetic means

Invariance of means (cites work)
cites work: Characterization of generalized quasi-arithmetic means

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cites work: Q4893576

Invariance of means (cites work)
cites work: Explicit solutions of the invariance equation for means

Invariance of means (cites work)
cites work: Q5622119

Invariance of means (cites work)
cites work: Q5620774

Invariance of means (cites work)
cites work: On the characterization of quasiarithmetic means with weight function

Invariance of means (cites work)
cites work: Q3786872

Invariance of means (cites work)
cites work: On the invariance equation for two-variable weighted nonsymmetric Bajraktarević means

Invariance of means (cites work)
cites work: A new estimate of the difference among quasi-arithmetic means

Invariance of means (cites work)
cites work: Limit properties in a family of quasi-arithmetic means

Invariance of means (cites work)
cites work: Iterated quasi-arithmetic mean-type mappings

Invariance of means (cites work)
cites work: Sur les moyennes

Invariance of means (cites work)
cites work: Q3669314

Invariance of means (cites work)
cites work: Generalizations of the Logarithmic Mean

Invariance of means (cites work)
cites work: Means in mathematical analysis. Bivariate means

Invariance of means (cites work)
cites work: Invariance in Some Families of Means

Invariance of means (cites work)
cites work: On invariance equation for means of power growth

Invariance of means (cites work)
cites work: An invariance of geometric mean with respect to generalized quasi-arithmetic means

Invariance of means (cites work)
cites work: On some invariance of the quotient mean with respect to Makó-Páles means

Invariance of means (DOI)
DOI: 10.1007/S00010-018-0564-5