Let $f$ be a $C^2$ real-valued function on the interval $[a,b]$. Suppose that $f'(x)$ is monotone on $[a,b]$ and there is $\lambda>0$ such that $$ \min_{x\in [a,b]} |f'(x)|>0. $$

It is not hard to see that there exists $C>0$ such that $$ \left|\int_a^b e^{2\pi i f(x)} dx \right|< \frac{C}{\lambda}. $$ The proof could be found, for instance, in the book uniform distribution of sequences Lemma 2.1.

I want to know whether we can find some upper bound respect to the length of $[a,b]$. More precisely, for the interval $[a,b]$ small enough (for example, small enough such that $|f(a)-f(b)|<1$), I wonder whether there exists a constant $C>0$ which is independent of $a, b$ and $f$ such that $$ \left|\int_a^b e^{2\pi i f(x)} dx \right|< \frac{C|a-b|}{\lambda}. $$

Let $f$ be a $C^2$ real-valued function on the interval $[a,b]$. Suppose that $f'(x)$ is monotone on $[a,b]$ and there is $\lambda>0$ such that $$ \min_{x\in [a,b]} |f'(x)|>\lambda $$

It is not hard to see that there exists $C>0$ such that $$ \left|\int_a^b e^{2\pi i f(x)} dx \right|< \frac{C}{\lambda}. $$ The proof could be found, for instance, in the book uniform distribution of sequences Lemma 2.1.

I want to know whether we can find some upper bound respect to the length of $[a,b]$. More precisely, for the interval $[a,b]$ small enough (for example, small enough such that $|f(a)-f(b)|<1$), I wonder whether there exists a constant $C>0$ which is independent of $a, b$ and $f$ such that $$ \left|\int_a^b e^{2\pi i f(x)} dx \right|< \frac{C|a-b|}{\lambda}. $$