Suppose that $\mathrm{f}:[\mathrm{a}, \mathrm{b}] \rightarrow \mathbb{R}$ is a smooth, convex function, and there exists a constant $\mathrm{t}>0$ such that $f^{\prime}(x) \geq t$ for all $x \in(a, b)$. Prove that $$ \left|\int_a^b e^{i f(x)} d x\right| \leq \frac{2}{t} $$
I tried to use some of the properties from the oscillatory integral and wrote the integrand as $f^{\prime}(x) e^{i f(x)} / f^{\prime}(x)$ but it seems there is some fundamental idea missing in my attempts.
Using integration by parts and noting that $f'(x) \geq t > 0$ and $f''(x) \geq 0$,
\begin{align*} \left| \int_{a}^{b} e^{if(x)} \, \mathrm{d}x \right| &= \left| \frac{e^{if(b)}}{if'(b)} - \frac{e^{if(a)}}{if'(a)} + \int_{a}^{b} \frac{f''(x)}{if'(x)^2}e^{if(x)} \, \mathrm{d}x \right| \\ &\leq \frac{1}{f'(b)} + \frac{1}{f'(a)} + \int_{a}^{b} \frac{f''(x)}{f'(x)^2} \, \mathrm{d}x \\ &\leq \frac{1}{f'(b)} + \frac{1}{f'(a)} + \left[ \frac{1}{f'(a)} - \frac{1}{f'(b)} \right] \\ &= \frac{2}{f'(a)} \leq \frac{2}{t}. \end{align*}
Remark. This bound cannot be improved further by noting that the equality holds if $f$ is linear with slope $t$ and $t(b - a)$ is an odd multiple of $\pi$.