Recently, I read Titchmarsh's writting "The Theory of the Riemann Zeta-Function" and there is a lemma in chapter 4 as follow:
Let $F(x)$ satisfy the conditions of the previous lemma, and let $G(x)/F'(x)$ be monotonic, and $|G(x)|\le M$. Then: $$\left|\int_b^a G(x)\cdot e^{iF(x)}\,\mathrm{d}x\right|\le\frac{8M}{\sqrt{r}}$$
Where $F(x)$ is a real function, twice differentiable, $F''(x)\geq r>0$ or $F''(x)\le r<0$ and throughout the interval $[a,b]$, and $G(x)$ is a real function. I wonder that if this lemma still holds when $\dfrac{G(x)}{F'(x)}$ is not monotonic.