  In this paper we show that a large class of one-dimensional Cohen-Macaulay
local rings $(A,\mathfrak{m})$ 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.
