On homogeneous quasideviation means

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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 x1, ⋯, xn∈ I then the equation E(x1, y) + ⋯ + E(xn, y) = 0 has a unique solution y = y0 which is between {Mathematical expression}xi and {Mathematical expression}xi (see [6]). This value y0 is called the E-quasideviation mean of x1, ⋯, xn and is denoted by {Mathematical expression}. The E-quasideviation mean {Mathematical expression} is called homogeneous if {Mathematical expression} is satisfied for all n ∈ ℕ, x1, ⋯, xn∈ I with tx1, ⋯, txn∈ 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 languageEnglish
Pages (from-to)132-152
Number of pages21
JournalAequationes Mathematicae
Volume36
Issue number2-3
DOIs
Publication statusPublished - 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

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