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 $1 < p < 2$.



Constant in the Marcinkiewicz-Zygmund inequality
D. Lubinsky in this [paper](http://www.ams.org/journals/proc/2014-142-10/S0002-9939-2014-12270-2/S0002-9939-2014-12270-2.pdf) shows that the best constant in the Marcinkiewicz-Zygmund inequality is equal to the best constant in the Polya-Plancherel inequality. This observation and an upper bound for the constant in R. P. Boas' book *Entire Functions* (Theorem 6.7.15 pg. 101) for the latter inequality may be useful. It appears the best constant is not known precisely ( see comments in Lubinsky's paper, page 2).


