On homogeneous quasideviation means

Summary

A functionE:I } I → ℝ is called aquasideviation on the open interval $I \subseteq \mathbb{R}_{\dot + }$ if

• (E1) sgnE(x, y) = sgn(x − y) forx, y ∈ I;

• (E2) y → E(x, y) is a continuous function onI for each fixedx ∈ I;

• (E3) y → E(x, y)/E(x′, y) is a strictly decreasing function on ]x, x′[ forx < x′ inI.

Ifx ₁,⋯, x _n ∈ I then the equation

E(x ₁,y) + ⋯ + E(x _n ,y) = 0 has a unique solutiony = 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 valuey ₀ is called theE-quasideviation mean ofx ₁,⋯, x _n and is denoted by$\mathfrak{M}_E (x_1 ,...,x_n )$.

TheE-quasideviation mean$\mathfrak{M}_E$ is calledhomogeneous if

$$\mathfrak{M}_E (tx_1 ,...,tx_n ) = t\mathfrak{M}_E (x_1 ,...,x_n )$$

is satisfied for alln ∈ ℕ,x ₁,⋯, x _n ∈ I withtx 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

• (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.