Let $\mu$ be a probability measure, $f$ a function taking values in $[0,1]$. I am trying to determine the sign of the expression
$3\left( \int f^2 d\mu \right)\left( \int f d\mu \right) - 2 \left( \int f d\mu \right)^3 -\left( \int f^3 d\mu \right).$
Under what conditions is this either non-positive or non-negative?