I am interested in a function $b : \mathbf{R}_+ \to [0, \infty]$ with the following properties:

• I know a priori that $b$ is a `totally monotone' function, i.e. for each nonnegative integer $k$, it holds that \begin{align} (-1)^k \left( \frac{\mathrm{d}}{\mathrm{d}t} \right)^{(k)} b \geq 0 \quad \text{on}\, \mathbf{R}_+\end{align}
• I am told that some function $\bar{b}$ is a global upper bound on $b$. A priori, I know relatively little about $\bar{b}$; let's say that I know that it is nonnegative and decreases to $0$.

I would like to argue that, given this information, I can form a new upper bound on $b$, say $\tilde{b}$ (again a function), such that

1. $\tilde{b}$ is always at least as good of an upper bound, i.e. $\tilde{b} \leq \bar{b}$, and,
2. $\tilde{b}$ is also a totally monotone function.

Essentially, I want to be able to say "WLOG, if I have an upper bound on a totally monotone function, then I can take that upper bound to be totally monotone".

Given that the set of totally monotone functions is a convex polytope, I am hopeful that there is a relatively simple argument which shows this (e.g. perhaps only using the convex / polytope structure), but I have not been able to crack it myself.