### Abstract

A function E:I } I → ℝ is called a quasideviation on the open interval {Mathematical expression} if (E1) sgn E(x, y) = sgn(x - y) for x, y ∈ I; (E2) y → E(x, y) is a continuous function on I for each fixed x ∈ I; (E3) y → E(x, y)/E(x′, y) is a strictly decreasing function on ]x, x′[ for x < x′ in I. If x_{1}, ⋯, x_{n}∈ I then the equation E(x_{1}, y) + ⋯ + E(x_{n}, y) = 0 has a unique solution y = y_{0} which is between {Mathematical expression}x_{i} and {Mathematical expression}x_{i} (see [6]). This value y_{0} is called the E-quasideviation mean of x_{1}, ⋯, x_{n} and is denoted by {Mathematical expression}. The E-quasideviation mean {Mathematical expression} is called homogeneous if {Mathematical expression} is satisfied for all n ∈ ℕ, x_{1}, ⋯, x_{n}∈ I with tx_{1}, ⋯, tx_{n}∈ I. One of the main results of the paper is the following Theorem. If I/I = {x/y {divides} x, y ∈ I} = ℝ_{+}and E:I } I → ℝ is an arbitrary function, then E is a quasideviation and {Mathematical expression}is a homogeneous mean if and only if there exist three functions a: I → ℝ_{+};f: ℝ_{+}→ ℝ, m:ℝ_{+}→ ℝ_{+}so that (i) a is continuous and positive; (ii) f is continuous and increasing on ℝ_{+}, further it is strictly monotonic on ]0, 1[ or on ]1, ∞[ and sgn f(x) = sgn(x - 1), x > 0; (iii) m is multiplicative, i.e. m(xy) = m(x)m(y) for x, y > 0; (iv) E(x, y) = a(y)m(x)f(x/y) for x, y ∈ I.

Original language | English |
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Pages (from-to) | 132-152 |

Number of pages | 21 |

Journal | Aequationes Mathematicae |

Volume | 36 |

Issue number | 2-3 |

DOIs | |

Publication status | Published - Jun 1 1988 |

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### Keywords

- AMS (1980) subject classification: Primary 39B99

### ASJC Scopus subject areas

- Mathematics(all)
- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Aequationes Mathematicae*,

*36*(2-3), 132-152. https://doi.org/10.1007/BF01836086