Bounds on the ratio between second raw moment and expected of absolute value squared

Question

I'm interested in bounds for the ratio $E[X^2]/E[|X|]^2$. The best lower bound is $1$ since $E[|X|] \geq E[X]$ and $E[X^2] - E[X]^2 = Var(X) \geq 0$. On the other hand, I would like to know if there is some upper bound or if we can find random variables with arbitrarily large ratios. Moreover, this quantity has a name?

Answer 1

There exists a r.v. $Y$ with $E|Y|<\infty$ but $EY^{2}=\infty$. Take $X_n=|Y|1_{|Y|\leq n}$ to see that the ratio can tend to $\infty$.

Explicit example of $Y$: Let $Y$ take values $1,2,3,...$ with $P(Y=n)=\frac c {n^{3}}$ where $c$ is chosen such that $\sum \frac c {n^{3}}=1$.