# Constant in the Marcinkiewicz-Zygmund inequality

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 $$1 < p < 2$$.