# Equality and homogeneity of generalized integral means

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## 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

$$M_{f,g,m;\mu}({\bf x}) :=\Bigl(\frac{f}{g}\Bigr)^{-1}\biggl( \frac{\int_{T} f(m(x_{1},\ldots,x_{d},t)){\rm d}\mu(t)} {\int_{T} g(m(x_{1},\ldots,x_{d},t)){\rm d}\mu(t)}\biggr) \quad ({\bf x} =(x_{1},\ldots,x_{d})\in I^{d}).$$

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

$$M_{f,g,m;\mu}( {\bf x} )=M_{h,k,m;\mu}( {\bf x} ) \quad ( {\bf x} \in I^{d})$$

and the homogeneity property

$$M_{f,g,m;\mu}(\lambda {\bf x} ) = \lambda M_{f,g,m;\mu}( {\bf x} ) \quad (\lambda>0,\, {\bf x} ,\lambda {\bf x} \in I^{d}),$$

respectively, be satisfied.

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## Author information

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Correspondence to Zs. Páles.

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