It is easy to show that if a real function $f:\mathbb{R}\rightarrow\mathbb{R}$ is contained in a strip $[a,b]$, that is if $\forall_{x}\, a\le f(x)\le b$, then its Laplace transform is bouned by $\frac{a}{s}$ from below and $\frac{b}{s}$ from above. The inverse is, however, not true, as one can find unbounded functions that have bounded Laplace transform.
My question is: are there any conditions which allow to verify if a function stays between given bounds knowing its Laplace transform? I am particularily interested in a class of rational functions with denominator being a polynomial of rank greater than 3 (such Laplace transform would desribe a bounded function if all the roots of the denominator lay on the left half of the complex plane, which can be checked by Routh-Hurwitz criterion, yet it does not provide the bounds of $f(x)$).