
> 
> **Question.** Let $X\_1,\dots,X\_n$ be random variables with normal distribution. Is it true that
>  $$\mathbb E \prod\_{i=1}^nX\_i^{2k}\ge\prod\_{i=1}^n\mathbb E X\_i^{2k}$$for any $k\in\mathbb N$?
> 
> 
> 


(The problem was posed on 22.06.2017 by Ph D students of H.Steinhaus Center of Wroclaw Polytechnica. The promised prize for a solution is "zywa kura", see pages [37,38](http://www.math.lviv.ua/szkocka/viewpage.php?vol=1&page=37) in the [Volume 1](http://www.math.lviv.ua/szkocka/viewbook.php?vol=1) of the [Lviv Scottish Book](http://www.math.lviv.ua/szkocka). To get the prize, write to the e-mail: hsc@pwr.edu.pl). 



An inequality for expected value of normally distributed variables
The problem posed above is a semi-well-known open problem that I believe is equivalent to the *real polarization conjecture*. A more general version of the question posed above is offered as Conjecture 4 in [this paper of Wenbo Li](https://link.springer.com/article/10.1007/s10959-010-0329-0), who considers arbitrary positive powers of the type $|X\_i|^{p\_i}$.


For the case $k=1$, a proof is known. Have a look at [Theorem 2.1](https://arxiv.org/pdf/0704.0028.pdf) in the linked paper for a proof of $k=1$ case (thanks to **Iosif** for alerting me that my answer was only about the $k=1$ case). In particular, the variables have to be jointly Gaussian with mean zero. Moreover, equality holds if and only if they are independent or at least one of them is a.s. zero.


