# On an equation involving weighted quasi-arithmetic means

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DOI: 10.1007/s10474-010-9246-z

- Cite this article as:
- Jarczyk, J. Acta Math Hung (2010) 129: 96. doi:10.1007/s10474-010-9246-z

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## Abstract

Let
which generalizes the Matkowski-Sutô equation. The paper completes a research stemming in the theory of invariant means.

*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$$
\kappa x + (1 - \kappa )y = \lambda \phi ^{ - 1} (\mu \phi (x) + (1 - \mu )\phi (y)) + (1 - \lambda )\psi ^{ - 1} (\nu \psi (x) + (1 - \nu )\psi (y))
$$

### Key words and phrases

meanfunctional equationquasi-arithmetic meaninvariant meanregularity of solutions### 2000 Mathematics Subject Classification

primary 26E60secondary 39B22## Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2010