Acta Mathematica Hungarica

, Volume 129, Issue 1, pp 96–111

On an equation involving weighted quasi-arithmetic means

Article

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

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.

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

Authors and Affiliations

  1. 1.Faculty of Mathematics, Computer Science and EconometricsUniversity of Zielona GóraZielona GóraPoland