Let $f$ and $g$ be two holomorphic functions in a connected open set $D$ of the plane, which have no zeros in $D$; if there is a sequence $(a_n)$ of points of $D$ such that
$\lim a_n = a, \qquad a \in D \quad\text{ and } a_n \neq a\text{ for all }n,$
and if \displaystyle\frac{f'(a_n)}{f(a_n)} = \frac{g'(a_n)}{g(a_n)}\qquad\text{ for all }n
show that there exists a constant $c$ such that $f(z) = c\cdot g(z)$ in $D$.
I've been penciling around with this question for a while and still can't get it. From what I can tell, the following theorems may be useful:
If $f$ is holomorphic in $D$ then $f$ is analytic in $D$ (can be represented as a power series)
If $\lim a_n = a$ for complex numbers $a_n$ and $a$ such that $a_n \neq a$ for all $n$, and $a$, $a_n$ all lie in $D$, then for an analytic function $f$ in $D$ and $f(a_n) = 0$ for all n, then $f = 0$ is the zero function.