relation between $|o(f)-g|$ and $|f-g|$

Question

This question is similar to the one asked some hours ago.

I have given three functions $f,g,h$ where $h(n)=o(f(n))$ and I know that $|f-g|<d<1$. Now I'd like to find an Expression for $|h-g|$.

Is there a General Expression?

I tried to do These calculations: $$|h-g|\leq |h-f|+d \leq |h|+|f|+d = |f|+d + o(|f|)$$ and $$|h-g|\leq |h|+|g| =O(d+g)$$

Is there something more we can say in this case?

Answer 1

No, there is not, since $h$ contains an unspecified constant. So, for instance, $f(n)=g(n)=n$ and $h(n)=100\sqrt{n}$ satisfy the assumptions.