Aequationes mathematicae

, Volume 92, Issue 5, pp 935–947 | Cite as

A note on functional equations connected with the Cauchy mean value theorem

  • Radosław Łukasik
Open Access


The aim of this paper is to describe the solution (fg) of the equation
$$\begin{aligned}{}[f(x)-f(y)]g'(\alpha x+(1-\alpha )y)= [g(x)-g(y)]f'(\alpha x+(1-\alpha )y),\ x,y\in I, \end{aligned}$$
where \(I\subset \mathbb {R}\) is an open interval, \(f,g:I\rightarrow \mathbb {R}\) are differentiable, \(\alpha \) is a fixed number from (0, 1).


Functional equation Mean value theorem Linearly dependent functions 

Mathematics Subject Classification



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Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of SilesiaKatowicePoland

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