Inequality with moments $m(f^3) \le m(f^2) m(f)$ Let $m$ a probability measure, $f$ a positive measurable function (one can assume it is bounded, the existence of the moments is not a problem here).
Is $m(f^3) \le m(f^2) m(f)$?

No. Actually, for every probability measure $m$ and nonnegative function $f$, $$m(f^3)\geqslant m(f^2)\cdot m(f),$$ with equality if and only if $f$ is ($m$-almost surely) constant.
Hence, checking any example would have shown that the conjecture is wrong.
