In mathematics, the Denjoy–Koksma inequality, introduced by Herman (1979, p.73) as a combination of work of Arnaud Denjoy and the Koksma–Hlawka inequality of Jurjen Ferdinand Koksma, is a bound for Weyl sums of functions f of bounded variation.

Suppose that a map f from the circle T to itself has irrational rotation number α, and p/q is a rational approximation to α with p and q coprime, |α – p/q| < 1/q². Suppose that φ is a function of bounded variation, and μ a probability measure on the circle invariant under f. Then

(Herman 1979, p.73)

• Herman, Michael-Robert (1979), "Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations", Publications Mathématiques de l'IHÉS (49): 5–233, ISSN 1618-1913, MR 0538680
• Kuipers, L.; Niederreiter, H. (1974), Uniform distribution of sequences, New York: Wiley-Interscience [John Wiley & Sons], ISBN 978-0-486-45019-3, MR 0419394, Reprinted by Dover 2006