Let $\mathcal{F}$ stand for the Fourier transform. Suppose $f : [-\delta/2,\delta/2] \to \mathbb{C}$ is a "nice" function. Is it true that $$\left|\mathcal{F} \left(e^{imx} \left(e^{ix^2}-1 \right)f(x) \right)(\xi) \right| \leq |\delta|^2 \left|\mathcal{F}\left(e^{imx}f(x) \right)(\xi)\right|?$$
If so, how can I prove it?
Thank you for your help.