Kusmin-Landau Inequality

Theorem
Let $I$ be the interval $(a,b]$.

Let $f:I\to R$ be continuously differentiable.

Suppose $f'$ is monotonic.

Suppose $\Vert f'\Vert\geq\lambda>0$ on $I$ for some $\lambda\in\R$, where $\Vert\cdot\Vert$ denotes distance to nearest integer.

Then $\displaystyle\sum_{n\in I}e^{2\pi if(n)}=O\left(\frac1\lambda\right)$, where the big-O estimate does not depend on $f$.