Aequationes mathematicae

, Volume 87, Issue 3, pp 301–308 | Cite as

On a functional equation related to competition

  • Peter KahligEmail author
  • Janusz Matkowski
Open Access


The functional equation
$$f \left(\frac{x + y}{1 - xy}\right) = \frac{f\left(x\right) + f\left(y\right)} {1 + f\left(x\right) f\left(y\right)}, \quad xy < 1,$$
(introduced by the first author in a competition model) is considered. The main result says that a function \({f : \mathbb{R} \rightarrow \mathbb{R}}\) satisfies this equation if, and only if, \({f = {\rm tanh} \circ \, \alpha \circ {\rm tan}^{-1}}\) , where \({\alpha : \mathbb{R} \rightarrow \mathbb{R}}\) is an additive function.

Mathematics Subject Classification (1991)

Primary 39B12 39B22 


Functional equation additive function general solution competition model 


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Copyright information

© The Author(s) 2013

Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations

  1. 1.ViennaAustria
  2. 2.Faculty of Mathematics Computer Science and EconometricsUniversity of Zielona GóraZielonaGóraPoland

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