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M-affine functions composing Sturm–Liouville families

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Abstract

Given an n-variable mean M defined on a real interval I, an M-affine function is a solution to the functional equation

When M is a quasilinear mean, the set of continuous M-affine functions is a Sturm–Liouville family on every compact interval \(\left[ a,b\right] \subseteq I\); i.e., for every \(\alpha ,\beta \in \left[ a,b\right] \), there exists an M-affine function f such that \(f\left( a\right) =\alpha \) and \( f\left( b\right) =\beta \). The validity of the converse statement is explored in this paper and several consequences are derived from this study. New characterizations of quasilinear means and the solution to Eq. (1) under suitable conditions are among the more important ones.

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Authors and Affiliations

  1. Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Laboratorio de Acústica y Electroacústica, Facultad de Cs. Exactas, Ing. y Agrim., Univ. Nac. de Rosario, Riobamba 245 Bis, 2000, Rosario, Argentina

    Lucio R. Berrone & Gerardo E. Sbérgamo

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  1. Lucio R. Berrone
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  2. Gerardo E. Sbérgamo
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Correspondence to Lucio R. Berrone.

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Berrone, L.R., Sbérgamo, G.E. M-affine functions composing Sturm–Liouville families. Aequat. Math. 92, 873–910 (2018). https://doi.org/10.1007/s00010-018-0588-x

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