# Inequality between two expectations!

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 $E|X|^j$ < $\infty$.

Hint: Apply Jensen inequality $c(\mathbb E(Y))\leqslant\mathbb E(c(Y))$ to the nonnegative random variable $Y=|X|^j$ and the convex function $c:\mathbb R_+\to\mathbb R$ defined by $c(t)=|t|^{k/j}$ for every $t$ in $\mathbb R_+$.