Holomorphic functions on unit disc Let $f,g$ be holomorphic on $\mathbb{D}:=\lbrace z\in\mathbb{C}:|z|<1\rbrace$, $f\neq0,g\neq0$, such that $$\frac{f^{\prime}}{f}(\frac{1}{n})=\frac{g^{\prime}}{g}(\frac{1}{n}) $$ for all natural $n\geq1$. Does it imply that $f=Cg$, where $C$ is some constant?
Let $A:=\lbrace\frac{1}{n}:n\geq1\rbrace$ and $h:=\frac{f^{\prime}}{f}-\frac{g^{\prime}}{g}$. Now, $h$ is holomprphic on $\mathbb{D}$ and disappears on a subset of $\mathbb{D}$ which has a limit point. Thus $h=0$, so $\frac{f^{\prime}}{f}=\frac{g^{\prime}}{g}$ on $\mathbb{D}$. 
Could someone help with the next steps? Or maybe $f$ doesn't have to be in the form described above?

Hint: Since $$\frac{f'}{f}=\frac{g'}{g}$$ on $\Bbb D$, then $f'g-fg'=0$ on $\Bbb D$, and so $$\left(\frac{f}{g}\right)'=\frac{f'g-fg'}{g^2}=0$$ on $\Bbb D$. Can you get the rest of the way from there and see why $\frac{f}{g}$ is holomorphic on $\Bbb D$?
