The [Marcinkiewicz–Zygmund inequality][1] states that

$$
{\displaystyle A_{p}E\left(\left(\sum _{i=1}^{n}\left\vert X_{i}\right\vert ^{2}\right)_{}^{p/2}\right)\leq E\left(\left\vert \sum _{i=1}^{n}X_{i}\right\vert ^{p}\right)\leq B_{p}E\left(\left(\sum _{i=1}^{n}\left\vert X_{i}\right\vert ^{2}\right)_{}^{p/2}\right)}.
$$
I wonder if there is any sort of result for the ratio of the two constant $A_p/B_p$, in particular for the case $p=1$.


[1]: https://en.wikipedia.org/wiki/Marcinkiewicz%E2%80%93Zygmund_inequality