Possible Duplicate:
Holomorphic functions and limits of a sequence
Hi There,
I was looking through an old text of mine, just refreshing myself on some material and I came across an interesting exercise statement that looks promising to understand. It just have been some time and I do not know if I am/even know how to approach the problem correctly. Some guidance of a solution for my thoughts to think on would be appreciated. The question is as stated:
Suppose that $f$ and $g$ are analytic on the disk $A=\{z \text{ such that }|z| \lt 2 \}$ and
that neither $f(z)$ nor $g(z)$ is ever $0$ for $z \in A.$ If
$ \frac{f^{\;'}(1/n)}{f(1/n)}=\frac{g'(1/n)}{g(1/n)} \quad {\text{ for }} \quad n=1,2,3,4 \ldots, $
could it be shown that there is a constant $c$ such that $f(z)=cg(z)~~\forall ~~ z \in A.$