Geodesic
Some logarithmic functional equations
Archivum mathematicum, Tome 48 (2012) no. 3, pp. 173-181 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
DOI : 10.5817/AM2012-3-173
Classification : 39B20
Keywords: logarithmic functional equation; Pexider equations 
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Laohakosol, Vichian; Pimsert, Watcharapon; Hengkrawit, Charinthip; Ebanks, Bruce. Some logarithmic functional equations. Archivum mathematicum, Tome 48 (2012) no. 3, pp. 173-181. doi: 10.5817/AM2012-3-173

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