I normally have problem with proving inequalities since there are many different inequalities and I'm usually confused on how to choose a proper one and focus on that to get my problem solved. Here is one of them that I thought Jensen's inequality should solve it but I've not been able to solve it yet.
Suppose $E|X|^k$ < $\infty$ show that for any j and k where $0 < j < k$ we have:
$$ (E|X|^j)^k \le (E|X|^k)^j $$
I think since $E|X|^k$ < $\infty$ and $ j < k $, then we can assume $E|X|^j$ < $\infty$.
I appreciate your hints.