## Abstract

Given two continuous functions
\(f,g \colon I \to\mathbb{R}\) such that *g* is positive
and *f/g* is strictly monotone, a measurable space \((T,\mathcal{A})\), a measurable family
of *d*-variable means \(m: I^{d} \times T \to I\), and a probability measure *μ* on the measurable
sets \(\mathcal{A}\), the *d*-variable mean \(M_{f,g,m;\mu} \colon I^{d} \to I\) is defined by

The aim of this paper is to solve the equality and homogeneity problems of these
means, i.e., to find conditions for the generating functions (*f, g*) and (*h, k*), for
the family of means *m*, and for the measure *μ* such that the equality

and the homogeneity property

respectively, be satisfied.

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The research of the first author was supported by the Hungarian Scientific Research Fund (OTKA) Grant K-111651 and by the EFOP-3.6.1-16-2016-00022, EFOP-3.6.2-16-2017-00015 projects. These projects are co-financed by the European Union and the European Social Fund.

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Páles, Z., Zakaria, A. Equality and homogeneity of generalized integral means.
*Acta Math. Hungar.* **160**, 412–443 (2020). https://doi.org/10.1007/s10474-019-01012-6

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DOI: https://doi.org/10.1007/s10474-019-01012-6