Let $f : \mathbb{R} / \mathbb{Z} \to \mathbb{C}$ be a trigonometric polynomial of degree $n$ and $m-1 \geq n$ be an integer. The Marcinkiewicz-Zygmund inequality asserts $$\int |f|^p \leq \frac{C_p}{m} \sum_{j=1}^m |f(j/m)|^p , \ \ \ 1 < p < \infty.$$

My question is known about the behavior $C_p$, as a function of $p$? I am particularly interested in the case $1 < p < 2$.