Let $X$ be a random variable with continuous density $\rho(x)$. Assume that $X$ is symmetric and $\vert X\vert
Is there anyone knows if the following statement is true or not? $ \frac{m_2}{2!}\times \frac{m_{4k}}{4k!}\geq \frac{m_{4k+2}}{(4k+2)!}. $ for $k\geq 1$. Note that one may rewrite the above equation as $ {4k+2\choose 2} m_2m_{4k}\geq m_{4k+2}. $ The Above recursion is true for some common distributions such as uniform distribution and Gaussian distribution (even though it does not have a bounded support) but can we say in general if it is true?
If not, what are the necessary conditions to make it true? For example, if $m_2>L^2/15$ then it is true. But is there any other condition available with less restriction?