Kusmin-Landau Inequality

Theorem
Let $I$ be the half-open interval $\hointl a b$.

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

Let $f'$ be monotonic.

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

Then:
 * $\ds \sum_{n \mathop \in I} e^{2 \pi i \map f n} = \map O {\frac 1 \lambda}$

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