# Article

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Keywords:
logarithmic functional equation; Pexider equations
Summary:
The functional equation $f(y-x) - g(xy) = h\left(1/x-1/y\right)$ is solved for general solution. The result is then applied to show that the three functional equations $f(xy)=f(x)+f(y)$, $f(y-x)-f(xy)=f(1/x-1/y)$ and $f(y-x)-f(x)-f(y)=f(1/x-1/y)$ are equivalent. Finally, twice differentiable solution functions of the functional equation $f(y-x) - g_1(x)-g_2(y) = h\left(1/x-1/y\right)$ are determined.
References:
[1] Chung, J.–Y.: A remark on a logarithmic functional equation. J. Math. Anal. Appl. 336 (2007), 745–748. DOI 10.1016/j.jmaa.2007.02.072 | MR 2348539 | Zbl 1130.39018
[2] Ebanks, B.: On Heuvers’ logarithmic functional equation. Result. Math. 42 (2002), 37–41. DOI 10.1007/BF03323552 | MR 1934223 | Zbl 1044.39018
[3] Heuvers, K. J.: Another logarithmic functional equation. Aequationes Math. 58 (1999), 260–264. DOI 10.1007/s000100050112 | MR 1715396
[4] Heuvers, K. J., Kannappan, P.: A third logarithmic functional equation and Pexider generalizations. Aequationes Math. 70 (2005), 117–121. DOI 10.1007/s00010-005-2792-8 | MR 2167989 | Zbl 1079.39019
[5] Kannappan, P.: Functional Equations and Inequalities with Applications. Springer, Dordrecht, 2009. MR 2524097 | Zbl 1178.39032
[6] Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities. second ed., Birkhäuser, Basel, 2009. MR 2467621 | Zbl 1221.39041
[7] Rätz, J.: On the theory of functional equation $f(xy) = f(x)+f(y)$. Elem. Math. 21 (1966), 10–13.

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