I came across the following inequality in one of my calculations ($X,Y$ are centered random variables):

$$\operatorname{E}(X^2Y^2)-\operatorname{E}(X^2)\operatorname{E}(Y^2) \geq 2 \operatorname{E}(XY)^2$$

or, written in terms of covariances,

$$\operatorname{Cov}(X^2,Y^2) \geq 2 \operatorname{Cov}(X,Y)^2$$.

If $(X,Y)=(U,V)$ is a two-dimensional centered Gaussian, this becomes an equality and if $(X,Y)=(H_p(U),H_q(V))$, where $(U,V)$ is still a two-dimensional centered Gaussian with $\operatorname{E}(U^2)=\operatorname{E}(V^2)=1$ and $H_k$ denotes the $k$th (probabilists') Hermite polynomial, the inequality above is strict whenever $p,q \geq 2$.

I have a feeling that something like this should be true for arbitrary random variables but couldn't prove it. Does this inequality look familiar to anybody or do you have an idea on how this could be proved/disproved?