### 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 language | English |
---|---|

Pages (from-to) | 157-218 |

Number of pages | 62 |

Journal | Publicationes Mathematicae |

Volume | 61 |

Issue number | 1-2 |

Publication status | Published - 2002 |

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### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Publicationes Mathematicae*,

*61*(1-2), 157-218.

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

Research output: Contribution to journal › Article

*Publicationes Mathematicae*, vol. 61, no. 1-2, pp. 157-218.

}

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 -