In this paper we show that a large class of one-dimensional Cohen–Macaulay local rings (\mathcal A,\mathfrak m) has the property that if M is a maximal Cohen–Macaulay A -module then the Hilbert function of M (with respect to \mathfrak m ) is non-decreasing. Examples include (1) complete intersections A = Q/(f,g) where (Q,\mathfrak n) is regular local of dimension three and f \in \mathfrak n^2 \setminus \mathfrak n^3 ; (2) one dimensional Cohen–Macaulay quotients of a two dimensional Cohen–Macaulay local ring with pseudo-rational singularity.