I am considering the problem of finding all functions $f:(0,\infty)\rightarrow(0,\infty)$ satisfying the functional equation:
$$f\big(xf(y)+f(x)\big) = 2f(x)+xy.$$
I have been able to prove the following three results/properties:
- $f$ is not surjective.
- $f$ does not have any fixed points.
- $f(x)=x+1$ is a solution.
My intuition tells me that the solution in (3) is the only solution, but I have not been successful in proving or disproving this claim.
Any ideas on how I can make further progress is appreciated.