Kusmin-Landau Inequality

Theorem
Let $I$ be the half-open interval $\left({a \,.\,.\, b}\right]$.

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

Let $f'$ be monotonic.

Let $\left\Vert{f'}\right\Vert \ge \lambda$ on $I$ for some $\lambda \in \R_{>0}$, where $\Vert\cdot\Vert$ denotes the distance to nearest integer.

Then:
 * $\displaystyle \sum_{n \mathop \in I} e^{2 \pi i f \left({n}\right)} = O \left({\frac 1 \lambda}\right)$

where the big-O estimate does not depend on $f$.