Suppose that

- $f$ and $g$ are polynomials with nonnegative coefficients,
- the degree of $g$ is greater than the degree of $f$,
- $g + f$ have no zeros on the right half plane $\mathbb{C}_+ = \{z \in \mathbb{C}\colon \Re z \ge 0 \}$.

Then $$ h = \frac{f}{g + f} $$ is a proper rational function that is analytic on $\mathbb{C}_+$. Hence, by the partial fraction decomposition, there exists $C > 0$ such that $$ |h^{(n)}(x)| \le \frac{Cn!}{x^{n+1}}, \qquad n \ge 0,\ x > 0.\tag{1} $$

Can we bound $C$ in terms of the coefficients of $f$ and $g$?

Or, in a more general setup:

Can we bound the coefficients of a partial fraction decomposition by a function of coefficients of a rational function?

My goal is to prove $(1)$ for fixed $C$, when coefficients of $f$ change. In particular:

Is it true that there exists $C > 0$ such that for every positive integer $k$ it follows that $$ \left|\left( \frac{kf}{g + kf} \right)^{(n)}(x)\right| \le \frac{Cn!}{x^{n+1}}, \qquad n \ge 0,\ x > 0? $$

This question is in a way a follow-up of Upper bound for the $n$-th derivative of a rational function $\frac{f}{f+g}$.