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Derivations and linear functions along rational functions

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Abstract

The main purpose of this paper is to give characterization theorems on derivations as well as on linear functions. Among others the following problem will be investigated: Let \({n \in \mathbb{Z}, f, g\colon\mathbb{R} \to\mathbb{R}}\) be additive functions, \({\left(\begin{array}{cc} a&b\\ c&d \end{array} \right) \in \mathbf{GL}_{2}(\mathbb{Q})}\) be arbitrarily fixed, and let us assume that the mapping

$$ \phi(x)=g\left(\frac{ax^{n}+b}{cx^{n}+d}\right)-\frac{x^{n-1}f(x)}{(cx^{n}+d)^{2}} \quad \left(x\in\mathbb{R}, cx^{n}+d\neq 0\right)$$

satisfies some regularity on its domain (e.g. (locally) boundedness, continuity, measurability). Is it true that in this case the above functions can be represented as a sum of a derivation and a linear function? Analogous statements ensuring linearity will also be presented.

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Authors and Affiliations

  1. Institute of Mathematics, University of Debrecen, P.O. Box: 12, Debrecen, 4010, Hungary

    Eszter Gselmann

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  1. Eszter Gselmann
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Correspondence to Eszter Gselmann.

Additional information

This research has been supported by the Hungarian Scientific Research Fund (OTKA) Grant NK 814 02 and by the TÁMOP 4.2.1./B-09/1/KONV-2010-0007 project implemented through the New Hungary Development Plan co-financed by the European Social Fund and the European Regional Development Fund.

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Gselmann, E. Derivations and linear functions along rational functions. Monatsh Math 169, 355–370 (2013). https://doi.org/10.1007/s00605-012-0375-z

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  • DOI: https://doi.org/10.1007/s00605-012-0375-z

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