Aequationes mathematicae

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

On a functional equation related to competition

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 


  1. 1.
    Kahlig P.: A model of competition. Appl. Math. 39, 293–303 (2012)MATHMathSciNetGoogle Scholar
  2. 2.
    Kahlig P.: Note to the paper “A model of competition”. Appl. Math. 40, 127 (2013)MATHMathSciNetGoogle Scholar
  3. 3.
    Matkowski, J.: The uniqueness of solutions of a system of functional equations in some classes of functions. Aequ. Math. 8, 233–237 (1972)Google Scholar
  4. 4.
    Kuczma, M.: An introduction to the theory of functional equations and inequalities. Cauchy’s equation and Jensen’s inequality. Uniwersytet Śla̧ski - PWN, Warszawa - Krakow - Katowice, 1985 (2nd edn., edited with a preface of Attila Gilányi, Birkhäuser verlag, Basel, 2009)Google Scholar

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

Personalised recommendations