On homogeneous quasideviation means (original) (raw)

Summary

A function_E:I } I_ → ℝ is called a_quasideviation on the open interval_ \(I \subseteq \mathbb{R}_{\dot + } \) if

  1. (E1)
    sgn_E(x, y)_ = sgn(x − y) for_x, y ∈ I_;
  2. (E2)
    y → E(x, y) is a continuous function on_I_ for each fixed_x ∈ I_;
  3. (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\(\mathop {min}\limits_{1 \leqslant i \leqslant n} \) x i and\(\mathop {max}\limits_{1 \leqslant i \leqslant n} \) x i (see [6]). This value_y_ 0 is called the_E-quasideviation mean_ of_x_ 1,⋯, x n and is denoted by\(\mathfrak{M}_E (x_1 ,...,x_n )\).

The_E_-quasideviation mean\(\mathfrak{M}_E \) is called_homogeneous_ if \mathfrak{M}_E (tx_1 ,...,tx_n ) = t\mathfrak{M}_E (x_1 ,...,x_n )$$

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 ∣ x, y ∈ I} = ℝ+ and E:I } I → ℝ is an arbitrary function, then E is a quasideviation and \(\mathfrak{M}_E \) is a homogeneous mean if and only if there exist three functions a: I → ℝ +;f: ℝ + → ℝ, m:ℝ + → ℝ + so that

  1. (i)
    a is continuous and positive;
  2. (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;
  3. (iii)
    m is multiplicative, i.e. m(xy) = m(x)m(y) for x, y > 0;
  4. (iv)
    E(x, y) = a(y)m(x)f(x/y) for x, y ∈ I.

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Authors and Affiliations

  1. Institute of Mathematics, Kossuth Lajos University, Pf. 12, H-4010, Debrecen, Hungary
    Zsolt Páles

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Páles, Z. On homogeneous quasideviation means.Aeq. Math. 36, 132–152 (1988). https://doi.org/10.1007/BF01836086

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