Consider the following: fix a function $\bar{b} : \mathbf{R}_+ \to [0, \infty]$, and define

\begin{align} \mathcal{S} \left( \bar{b} \right) := \left\{ b : \mathbf{R}_+ \to [0, \infty] \, \text{s.t.} \, b \leq \bar{b} \, \text{pointwise} \right\}. \end{align}

Recall the set of totally-monotone functions $\mathcal{B}$, defined as the set of functions $b$ so that for each nonnegative integer $k$, it holds that \begin{align} \text{for} \, t \in \mathbf{R}_+, \quad (-1)^k \left( \frac{\mathrm{d}}{\mathrm{d}t} \right)^{(k)} b \geq 0 \quad \end{align}

Treating $\bar{b}$ as fixed, I would like to find a totally-monotone function $\tilde{b}$ which is a valid upper bound for all functions in $\mathcal{S} \left( \bar{b} \right) \cap \mathcal{B}$.

That is, given the set of totally-monotone functions which are upper-bounded by $\bar{b}$, I would like to be able to say that the same functions can also be upper-bounded by a totally-monotone function $\tilde{b}$.

Ideally, it would also be the case that this new bound is at least as tight as the original bound, i.e.

\begin{align} 0 \leq \tilde{b} \leq \bar{b} \end{align}

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.