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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,

$$ \frac{f(\frac{x+y}{2})}{g(\frac{x+y}{2})} = \frac{\frac{f(x)}{g(x)} + \frac{f(y)}{g(y)}}{2} $$

Adrián Álvarez-Vázquez
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    What is your question, really? f(x) anf g(x) are any two such functions that their ratio is a solution to the Cauchy functional equation, which is pretty well studied. – Ivan Neretin Sep 14 '17 at 11:51
  • My question is about the set of the class functions $f$ and $g$ that satisfy that equation. I am novel in the functional equations field (just versed with Lectures of Aczél and Kuczma), but I imagine a certain set of functions (huge maybe) is posible to be defined for functions $f$ and $g$ with a linear coefficient between them. – Adrián Álvarez-Vázquez Sep 14 '17 at 12:11
  • The problem comes from an experimental relation between two variables, designated here as $f$ and $g$ functions, such its quotient is linear. Then, can be stated this quotient satisfy Cauchy functional equation. Consequently, I wish to know what kind of functions can be $f$ and $g$, from functional equations theory, such that satisfy these conditions. – Adrián Álvarez-Vázquez Sep 14 '17 at 12:25
  • Functional equation has nothing to do with this. It was solved long ago. Your representation of the unknown function as f/g does not add anything new to it. You already know what is f/g, now you want f and g. – Ivan Neretin Sep 14 '17 at 12:39
  • I suppose a possible generalization of the Cauchy functional equation (with quotient of functions instead of functions themselves) could be possible to be stated. In this sense, I can imagine certain functions (as polynomial/polynomial, etc.) such a linear quotient, but I attend a generalized and formalized definition of that set of functions, maybe through the functional equations theory. – Adrián Álvarez-Vázquez Sep 14 '17 at 12:47
  • Your generalization does not generalize anything at all. – Ivan Neretin Sep 14 '17 at 13:32

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