A third logarithmic functional equation and Pexider generalizations

Summary.

Let $f:]0,\infty [ \to {\mathbb{R}}$ be a real valued function on the set of positive reals. Then the functional equations:

$$\begin{aligned} f(x + y) - f(xy) = & f(1/x + 1/y) \\ f(x + y) - f(x) - f(x) = & f(1/x + 1/y) \\ \end{aligned} $$

and

$$f(xy) = f(x) + f(y)$$

are equivalent to each other.

If $f,g,h:]0,\infty [ \to {\mathbb{R}}$ are real valued functions on the set of positive reals then

$$f(x + y) - g(xy) = h(1/x + 1/y)$$

is the Pexider generalization of

$$f(x + y) - f(xy) = f(1/x + 1/y).$$

We find the general solution to this Pexider equation.

If $f,g,h,k:]0,\infty [ \to {\mathbb{R}}$ are real valued functions on the set of positive reals then

$$f(x + y) - g(x) - h(y) = k(1/x + 1/y)$$

is the Pexider generalization of

$$f(x + y) - f(x) - f(y) = f(1/x + 1/y).$$

We find the twice differentiable solution to this Pexider equation.