Inequality between $\mathbb{E}[XY]$ and $\mathbb{E}[X^2]$ if $X$ and $Y$ have the same distribution Does there exist an inequality connecting  $\mathbb{E}[XY]$ and $\mathbb{E}[X^2]$ if $X$ and $Y$ have the same distribution, regardless of whether they are independent or not?

Note that $0\le E(X-cY)^2=c^2EY^2-2cEXY+EX^2$ for all $c\in\Bbb R$, so this quadratic in $c$ has discriminant $\le0$, i.e. $4c^2(EXY)^2-4c^2EX^2EY^2\le0$. This simplifies to $|EXY|\le EX^2EY^2$.
