Zeros of holomorphic function have limit point outside domain The following is exercise 10.20 in Rudin's R&CA.

Suppose $f \in H(U)$, $g \in H(U)$, and neither $f$ nor $g$ has a zero in $U$. If
  $$
\frac{f'}{f}\left(\frac{1}{n}\right) = \frac{g'}{g}\left(\frac{1}{n}\right) \quad (n = 1, 2, 3, \ldots)
$$
Find another simple relation between $f$ and $g$.

EDIT: I didn't know $U$ is the unit disc when I posted this. I thought it could be any open set.
$0$ is a limit point of the zeros of the function $\varphi = f'/f - g'/g$. However, $0$ isn't necessarily in $U$. I'm not sure how to deal with it if $0$ is a pole or essential singularity of $\varphi$, and I couldn't find much info about limit points of zeros that are outside the domain of a holomorphic function. Ultimately, I suspect $\varphi = 0$ everywhere and thus $g = \lambda f$ for some constant $\lambda$.
Am I on the right track? Is there a better approach here? Thanks.

Let $h:=f/g$ (well defined). Its derivative is $0$ and if $U$ is the open unit disk, we get $h'\equiv 0$ hence $f$ and $g$ are proportional. 
