Gauss-composition of means and the solution of the Matkowski-Sutô problem

Zoltán Daróczy, Z. Páles

Research output: Contribution to journalArticle

63 Citations (Scopus)

Abstract

In this paper the so-called Matkowski-Sutô problem is completely solved, that is, continuous and strictly monotonic functions φ and ψ defined on an open real interval I are determined such that the functional equation (1) φ-1 (φ(x) + φ(y)/2) + ψ-1 (ψ(x) + ψ(y)/2) = x + y holds for all x, y ∈ I. The above equation belongs to the class of so-called composite functional equations that does not possess a regularity theory such as known for non-composite equations. The main results of the paper offer new methods to obtain higher-order regularity properties of the unknown functions φ and ψ. First, based on Lebesgue's theorem on the almost everywhere differentiability of monotonic functions, the local Lipschitz property of φ and ψ and their inverses is shown. Then the differentiability of these functions is proved in a subinterval of I. Finally, using Baire's theorem on the continuity properties of derivative functions, the continuous differentiability of φ and ψ in a subinterval is deduced. After these regularity properties, the equation is solved in the subinterval so obtained with earlier methods of the authors. The proof is then completed by using the extension theorem due to the authors and Gy. Maksa. The main result obtained is a generalization of that of Sutô (1914) and Matkowski (1999). As application, the connection to Gauss composition of means, the equality problem of quasi-arithmetic means and conjugate arithmetic means is discussed and solved.

Original languageEnglish
Pages (from-to)157-218
Number of pages62
JournalPublicationes Mathematicae
Volume61
Issue number1-2
Publication statusPublished - 2002

Fingerprint

Differentiability
Gauss
Monotonic Function
Regularity Properties
Functional equation
Quasi-arithmetic Mean
Lipschitz Property
Regularity Theory
Extension Theorem
Local Properties
Henri Léon Lebésgue
Theorem
Equality
Strictly
Composite
Higher Order
Derivative
Unknown
Interval

Keywords

  • Composition of means
  • Gauss-iteration
  • Invariance equation
  • Quasi-arithmetic mean

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Gauss-composition of means and the solution of the Matkowski-Sutô problem. / Daróczy, Zoltán; Páles, Z.

In: Publicationes Mathematicae, Vol. 61, No. 1-2, 2002, p. 157-218.

Research output: Contribution to journalArticle

@article{09fe99f4032446bf917b6641cdf6d4d5,
title = "Gauss-composition of means and the solution of the Matkowski-Sut{\^o} problem",
abstract = "In this paper the so-called Matkowski-Sut{\^o} problem is completely solved, that is, continuous and strictly monotonic functions φ and ψ defined on an open real interval I are determined such that the functional equation (1) φ-1 (φ(x) + φ(y)/2) + ψ-1 (ψ(x) + ψ(y)/2) = x + y holds for all x, y ∈ I. The above equation belongs to the class of so-called composite functional equations that does not possess a regularity theory such as known for non-composite equations. The main results of the paper offer new methods to obtain higher-order regularity properties of the unknown functions φ and ψ. First, based on Lebesgue's theorem on the almost everywhere differentiability of monotonic functions, the local Lipschitz property of φ and ψ and their inverses is shown. Then the differentiability of these functions is proved in a subinterval of I. Finally, using Baire's theorem on the continuity properties of derivative functions, the continuous differentiability of φ and ψ in a subinterval is deduced. After these regularity properties, the equation is solved in the subinterval so obtained with earlier methods of the authors. The proof is then completed by using the extension theorem due to the authors and Gy. Maksa. The main result obtained is a generalization of that of Sut{\^o} (1914) and Matkowski (1999). As application, the connection to Gauss composition of means, the equality problem of quasi-arithmetic means and conjugate arithmetic means is discussed and solved.",
keywords = "Composition of means, Gauss-iteration, Invariance equation, Quasi-arithmetic mean",
author = "Zolt{\'a}n Dar{\'o}czy and Z. P{\'a}les",
year = "2002",
language = "English",
volume = "61",
pages = "157--218",
journal = "Publicationes Mathematicae",
issn = "0033-3883",
publisher = "Kossuth Lajos Tudomanyegyetem",
number = "1-2",

}

TY - JOUR

T1 - Gauss-composition of means and the solution of the Matkowski-Sutô problem

AU - Daróczy, Zoltán

AU - Páles, Z.

PY - 2002

Y1 - 2002

N2 - In this paper the so-called Matkowski-Sutô problem is completely solved, that is, continuous and strictly monotonic functions φ and ψ defined on an open real interval I are determined such that the functional equation (1) φ-1 (φ(x) + φ(y)/2) + ψ-1 (ψ(x) + ψ(y)/2) = x + y holds for all x, y ∈ I. The above equation belongs to the class of so-called composite functional equations that does not possess a regularity theory such as known for non-composite equations. The main results of the paper offer new methods to obtain higher-order regularity properties of the unknown functions φ and ψ. First, based on Lebesgue's theorem on the almost everywhere differentiability of monotonic functions, the local Lipschitz property of φ and ψ and their inverses is shown. Then the differentiability of these functions is proved in a subinterval of I. Finally, using Baire's theorem on the continuity properties of derivative functions, the continuous differentiability of φ and ψ in a subinterval is deduced. After these regularity properties, the equation is solved in the subinterval so obtained with earlier methods of the authors. The proof is then completed by using the extension theorem due to the authors and Gy. Maksa. The main result obtained is a generalization of that of Sutô (1914) and Matkowski (1999). As application, the connection to Gauss composition of means, the equality problem of quasi-arithmetic means and conjugate arithmetic means is discussed and solved.

AB - In this paper the so-called Matkowski-Sutô problem is completely solved, that is, continuous and strictly monotonic functions φ and ψ defined on an open real interval I are determined such that the functional equation (1) φ-1 (φ(x) + φ(y)/2) + ψ-1 (ψ(x) + ψ(y)/2) = x + y holds for all x, y ∈ I. The above equation belongs to the class of so-called composite functional equations that does not possess a regularity theory such as known for non-composite equations. The main results of the paper offer new methods to obtain higher-order regularity properties of the unknown functions φ and ψ. First, based on Lebesgue's theorem on the almost everywhere differentiability of monotonic functions, the local Lipschitz property of φ and ψ and their inverses is shown. Then the differentiability of these functions is proved in a subinterval of I. Finally, using Baire's theorem on the continuity properties of derivative functions, the continuous differentiability of φ and ψ in a subinterval is deduced. After these regularity properties, the equation is solved in the subinterval so obtained with earlier methods of the authors. The proof is then completed by using the extension theorem due to the authors and Gy. Maksa. The main result obtained is a generalization of that of Sutô (1914) and Matkowski (1999). As application, the connection to Gauss composition of means, the equality problem of quasi-arithmetic means and conjugate arithmetic means is discussed and solved.

KW - Composition of means

KW - Gauss-iteration

KW - Invariance equation

KW - Quasi-arithmetic mean

UR - http://www.scopus.com/inward/record.url?scp=0036376868&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036376868&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0036376868

VL - 61

SP - 157

EP - 218

JO - Publicationes Mathematicae

JF - Publicationes Mathematicae

SN - 0033-3883

IS - 1-2

ER -