Is it possible to prove the below inequality: $$ \left\lvert{\int_{-\varepsilon}^\varepsilon e^{-xt^2}\left(e^{-xR(t)}-1\right)\mathrm{d}{t}}\right\rvert \leq C\left\lvert x\int_{-\varepsilon}^\varepsilon e^{-xt^2}\lvert{t}\rvert^3\mathrm{d}{t}\right\rvert \qquad\text{where } x\in \mathbb{C}, \lvert x\rvert \gg 0, \Re(x)\geq 0 $$ Here, $R(t)$ is $O(t^3)$ and we have the freedom to pick any $\varepsilon >0$.
Thanks in advance!
My idea: The inequality is trivial if $x$ is real. So for the complex case, I want $\varepsilon$ to be small so that $e^{-xt^2}$ is "almost" real. I am looking for a way to write this out rigorously.