I know that a certain function $z(x) = f(x)/g(x)$ exhibit a linear behavior:
$$ z(x) = \frac{f(x)}{g(x)} = Ax+B $$ where A and B are constants. That is, can be assumed $z(x)$ satisfies the Cauchy functional equation in a special form,
$$ \frac{f(x+y)}{g(x+y)} + B = \frac{f(x)}{g(x)} + \frac{f(y)}{g(y)} $$
where B is an arbitrary constant. I wish to know the set of functions $f$ and $g$ that satisfy the above functional equation (not least what is possible to know about them), that is, its quotient exhibit a linear behavior (which is the same that satisfy the Cauchy functional equation).
Then, by letting $x= 0 $,
$$ \frac{f(0)}{g(0)} = B $$
the same result is obtained if $y=0$.
Through the original Cauchy proceeding for the solution by induction of the functional equation, that is, assuming $x= y$,
$$ \frac{f(nx)}{g(nx)} = n \frac{f(x)}{g(x)} + B $$
Maybe $z(x)$ could be assumed satisfying Jensen's functional equation, since its linear behavior, that is,
$$ \frac{f(\frac{x+y}{2})}{g(\frac{x+y}{2})} = \frac{\frac{f(x)}{g(x)} + \frac{f(y)}{g(y)}}{2} $$