I've been trying to prove: $m_4m_2\geq m_3^2+m_2^2$
How I proceeded:
From the Cauchy-Schwarz inequality:
$m_3=E[(x-\bar{x})^3]$
$m_2=E[(x-\bar{x})^2]$
$m_4=E[(x-\bar{x})^4]$
$[E(XY)]^2 \leq E[X^2] \cdot E[Y^2]$
So, $m_3^2=[E(x-\bar{x})(x-\bar{x})^2]^2 \leq E[(x-\bar{x})^2] \cdot E(x-\bar{x})^4]$
But this gives $m_3^2 \leq m_4m_2$
Here, $m_r$ are central moments
This isn't what I wanted. Am I in the wrong here?