I have two functions $f(t)$, $g(t)$ defined by their Laplace Transforms $F(s)$ and $G(s)$ and I want to bound ratio $f(t)/g(t)$. Is it possible to by looking at $F(s)$, $G(s)$ or $\mathcal{L}^{-1}(F/G)$?
The issue is that I can only invert one of the transforms, or their ratio:
$$G(s)=\frac{\tan ^{-1}\left(\frac{\sqrt{2}}{\sqrt{s}}\right)}{\sqrt{2} \sqrt{s} \left(\frac{1}{4} \left(\sqrt{2} \sqrt{s} \tan ^{-1}\left(\frac{\sqrt{2}}{\sqrt{s}}\right)-2\right)+1\right)}$$
$$F(s)=\frac{\tan ^{-1}\left(\frac{\sqrt{2}}{\sqrt{s}}\right)}{\sqrt{2} \sqrt{s}}$$
$$f(t)=\mathcal{L}^{-1}(F)=\frac{\sqrt{\frac{\pi }{2}} \text{erf}\left(\sqrt{2} \sqrt{t}\right)}{2 \sqrt{t}}$$
$$\mathcal{L}^{-1}(F/G)=\frac{\delta (t)}{2}+\frac{1}{16} \left(\frac{4 e^{-2 t}}{t}-\frac{\sqrt{2 \pi } \text{erf}\left(\sqrt{2} \sqrt{t}\right)}{t^{3/2}}\right)$$
I can numerically invert it to see that two functions are fairly close to each other, but need a more rigorous proof.
