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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
Article
  • 36 Downloads

Abstract

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).

Keywords

Functional equation Mean value theorem Linearly dependent functions 

Mathematics Subject Classification

39B22 

References

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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 (http://creativecommons.org/licenses/by/4.0/), 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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