, Volume 129, Issue 1-2, pp 96-111
Date: 08 May 2010

On an equation involving weighted quasi-arithmetic means

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Abstract

Let I ⊂ ℝ be an interval and κ, λ ∈ ℝ / {0, 1}, µ, ν ∈ (0, 1). We find all pairs (φ, ψ) of continuous and strictly monotonic functions mapping I into ℝ and satisfying the functional equation

$$ \kappa x + (1 - \kappa )y = \lambda \phi ^{ - 1} (\mu \phi (x) + (1 - \mu )\phi (y)) + (1 - \lambda )\psi ^{ - 1} (\nu \psi (x) + (1 - \nu )\psi (y)) $$
which generalizes the Matkowski-Sutô equation. The paper completes a research stemming in the theory of invariant means.