Sufficient condition for the characteristic function of a discrete distribution to be decreasing

Question

Let $X$ be a discrete random variable defined on the lattice such that $P(X \in \mathbb{Z}/h) = 1$, where $\mathbb{Z}/h$ represents the set of integers scaled by a factor of $h$. I wonder if there exists a sufficient condition for the characteristic function of $X$ to be decreasing over the interval $[0, \pi h]$. Also, one may assume the probability density of $X$ is symmetric.

Thanks!!!