Let $f$ and $g$ be real nonzero polynomials with nonnegative coefficients such that $g(0) \neq 0$.
Let $$ h(x) = \frac{xf(x)}{(x^2+x)f(x) + g(x)}. $$ In particular, it follows that $h$ is well defined on the interval $[0,+\infty)$, and $$ 0 \le h(x) \le 1, \qquad x \ge 0. $$ Even more, using the partial fractions decomposition, we have that there is a constant $C > 0$ such that the $n$-th derivative of $h$ satisfies $$ |h^{(n)}(x)| \le C \frac{n!}{x^{n+1}}, \qquad x > 0,\ n \in \mathbb{N}. $$
Is it true that we can choose $C$ as above that does not depend on the coefficients of polynomials $f$ and $g$?
If this is not possible, are there some "natural" conditions that guarantee the existence of such $C$?
Note: This is a follow-up question to this.