Consider a real random variable $X$ with zero mean. Does the following inequality hold in general?
$$\langle X^4\rangle \ge 3 \langle X^2\rangle^2$$
I'm not sure how to prove this or if a counter-example exists. If the inequality is true, it is also tight because for a standard normal this is an equality.
I found a related inequality, valid for symmetric distributions (all odd moments vanish):
$$\langle X^4\rangle \ge 2\langle X^2\rangle$$
This is proved by
Dreier, Ilona. "Inequalities between the second and fourth moments." Statistics: A Journal of Theoretical and Applied Statistics 32.2 (1998): 189-198.
But I'm actually not sure if this is connected to the inequality above.
Update: The inequality from Dreier 1998 is stated under the assumption of a distribution with a non-negative characteristic function.