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I want to compare the quantities $(E_X[X^2])^6$ and $E_X[X^{12}]$.

So, $g(X)=X^2$ is strictly convex since $g''(X)=(X^2)''=2>0$ and thus $$E_X[X^2]\geq(E_X[X])^2\Rightarrow(E_X[X^2])^6\geq(E_X[X])^{12}$$ Now we have to check whether $(E_X[X])^{12}\geq E_X[X^{12}]$. But $g(X)=X^{12}$ is convex since $g''(X)=(X^{12})''=132X^{10}\geq 0$. As a result, I cannot proceed with this inequality.

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    How about $g(X)=X^6$? – 2017-01-13

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The function $f(x) = x^6$ is convex, so applying Jensen to the random variable $X^2$ gives $$E(X^2)^6 \le E((X^2)^6) = E(X^{12})$$