In my research (I am not a mathematician by training, but I frequently use mathematics in my research), I often come across "approximation" problems in the following form:

Let $X$ and $Y$ be two possibly dependent continuous non-negative real-valued random variables. One can assume all the nice things about the PDFs of $X$ and $Y$. The situation is now as follows: Usually, I can prove that an estimate of the form $X \leq Y \leq \beta X$ for some $\beta > 1$ holds. The ultimate goal is to find an upper bound on the integral $I = \int_{a}^{a(1+\epsilon)} f_Y(y)\mathrm{d}y$, where $a>0$ and $\epsilon>0$ is "small." Here, the assumption is that the PDF of $Y$ is "extremely hard to evaluate" and one hopes to find an upper bound using the PDF of $X$ (which is much easier to evaluate let's say) only.

One can go by writing $I = F_Y(a(1+\epsilon)) - F_Y(a)$ and bound each term via the CDF of $X$. Suppose that by doing so, we get the upper bound $I_{\mathrm{bad}}$ on $I$. The problem is that obviously, $I_{\mathrm{bad}}$ will not decay to $0$ as $\epsilon \rightarrow 0$ while $I$ does. I was wondering whether there is another trick of the trade that will give a more sensical bound for small $\epsilon$. I am not (necessarily) looking for an answer that will hold for any distribution; even ideas or methods regarding a specific class of distributions would be greatly appreciated. Thanks for your time.