Which of the following inequalities are $\textbf{not always}$ true: $$(1). E|X| \geq |EX|$$ $$(2). E[X^4] \geq \frac{(EX)^4}{E[X^6]} \geq E[X]^6$$ $$(3).E[e^X] \geq E(1 + \frac{X}{1!} + \frac{X^2}{2!}) $$ $$(4). E(X^2 + \frac{1}{X^2} + e^{3X}) \geq E[3e^X]$$
$(1)$ is directily given by Jensen's inequality. (3) can be obtained by Taylor expansion : $e^X = \sum_{k=0}^{\infty}\frac{X^k}{k!} \geq 1 + \frac{X}{1!} + \frac{X^2}{2!}$.
For (2), I tried to use Holder's inequality to show $$(EX)^4 \leq E[X^4]E[X^6]$$ but couldn't really figure it out yet, and I have no idea how to approach (4).
Any ideas or hints are welcome. Thanks all in advance.
(2) is not true, for example you can let $X=10$
(4) is true, you can use AM-GM inequality, $a^3+b^3+c^3\ge 3abc$