logarithmic functional equation; Pexider equations
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.
 Kannappan, P.: Functional Equations and Inequalities with Applications
. Springer, Dordrecht, 2009. MR 2524097
| Zbl 1178.39032
 Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities
. second ed., Birkhäuser, Basel, 2009. MR 2467621
| Zbl 1221.39041
 Rätz, J.: On the theory of functional equation $f(xy) = f(x)+f(y)$. Elem. Math. 21 (1966), 10–13.