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Some logarithmic functional equations

The functional equation f ( y - x ) - g ( x y ) = h 1 / x - 1 / y is solved for general solution. The result is then applied to show that the three functional equations f ( x y ) = f ( x ) + f ( y ) , f ( y - x ) - f ( x y ) = 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 1 / x - 1 / y are determined.

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