Suppose the random variables $X_i$ are independent and satisfy $E[X_i] = 0$. Then the following inequality holds: $$E\left[\left(\sum \limits_{i = 1}^n X_i\right)^4\right] = \sum \limits_{i = 1}^n E[X_i^4] + 3\sum \limits_{i \ne j} E[X_i^2X_j^2] \le 3\left(\sum \limits_{i = 1}^n E[X_i^4] + \sum \limits_{i \ne j} \sqrt{E[X_i^4]}\sqrt{E[X_j^4]}\right) = 3\left(\sum \limits_{i = 1}^n \sqrt{E[X_i^4]}\right)^2$$
Now I would like to generalize this inequality to moments of order 6 [or higher]. My best guess is that the inequality
$$E\left[\left(\sum \limits_{i = 1}^n X_i\right)^6\right] \le C \left(\sum \limits_{i = 1}^n E[X_i^6]^{1/3}\right)^3$$
holds, where $C$ is an absolute constant.
Can someone provide a proof/ source for the last inequality? Or perhaps a source for a similar inequality?
The "similar inequality" should provide at least $E\left[\left(\sum \limits_{i = 1}^n X_i\right)^6\right] = O(n^3)$ in the i.i.d. case.
The inequality is actually correct. It follows from Theorems 3 in a Rosenthal paper from 1970 (Israel J. Math. 8, 273-303).
Rosenthal proves the following inequality for $p > 2$ and independent, centered Random variables $X_i \in L^p$: $$E\left[\left|\sum \limits_{i = 1}^n X_i\right|^p\right] \le C(p) \max\left\{\sum \limits_{i = 1}^n E[ \left|X_i\right|^p], \left(\sum \limits_{i = 1}^n E[X_i^2]\right)^{p/2}\right\}.$$
The inequality in my original post follows from two applications of Jensen's inequality: $$\left(\sum \limits_{i = 1}^n E[\left|X_i\right|^p]\right)^{2/p} = n^{2/p} \left(\sum \limits_{i = 1}^n \frac{1}{n}E[\left|X_i\right|^p]\right)^{2/p} \le n^{2/p - 1} \sum \limits_{i = 1}^n E[\left|X_i\right|^p]^{2/p} \le \sum \limits_{i = 1}^n E[\left|X_i\right|^p]^{2/p}$$ $$E[X_i^2] \le E[\left|X_i\right|^p]^{2/p}$$