Relation of $\frac{f}{g}$ to $\frac{f'}{f}-\frac{g'}{g}$ in reverse order.

Suppose $f$ and $g$ have following properties: $f(0)=g(0)=0$, $f(1)=g(1)=1$, $f'(0)=g'(0)=0$, $f''(x)>0, f'''(x)>0, g''(x)>0, g'''(x)>0 \ \forall x\in(0,1)$.

Then I want to show that $f(x)>g(x) \ \forall x\in(0,1) \Rightarrow \frac{f'(x)}{f(x)}<\frac{g'(x)}{g(x)} \ \forall x\in(0,1)$, or to find a counterexample

Alternatively, RHS above simply states that $\frac{f}{g}$ is decreasing on $(0,1)$

This is trivially true in reverse order i.e $\frac{f'(x)}{f(x)}<\frac{g'(x)}{g(x)} \ \forall x\in(0,1) \Rightarrow f(x)>g(x) \ \forall x\in(0,1)$