## Abstract

The purpose of this paper is to investigate the following invariance equation involving two 2-variable generalized Bajraktarević means, i.e., we aim to solve the functional equation

where *I* is a nonempty open real interval and \(f,g \colon I \to\mathbb{R}\) are continuous, strictly monotone and \(p_1,p_2,q_1,q_2 \colon I \to\mathbb{R}_+\) are unknown functions.
The main result of the paper shows that, assuming four times continuous differentiability of *f*, *g*, twice continuous differentiability of *p*_{1} and *p*_{2} and assuming that *p*_{1} differs from *p*_{2} on a dense subset of *I*, a necessary and sufficient condition for the equality above is that the unknown functions are of the form

where \(u,v,w,z \colon I \to\mathbb{R}\) are arbitrary solutions of the second-order linear differential equation \(F''=\gamma F (\gamma\in\mathbb{R}\) is arbitrarily fixed) such that *v* > 0 and *z* > 0 holds on *I* and \(\{u,v\}\) and \(\{w,z\}\) are linearly independent.

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The research of the first author was supported by the ÚNKP-20-3 New National Excellence Program of the Ministry of Human Capacities.

The research of the second author was supported by the K-134191 NKFIH Grant and the 2019-2.1.11-TÉT-2019-00049 and the EFOP-3.6.1-16-2016-00022 projects. The last project is cofinanced by the European Union and the European Social Fund.

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Grünwald, R., Páles, Z. On the invariance of the arithmetic mean with respect to generalized Bajraktarević means.
*Acta Math. Hungar.* **166**, 594–613 (2022). https://doi.org/10.1007/s10474-022-01230-5

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DOI: https://doi.org/10.1007/s10474-022-01230-5