Inequality for inverse random variable

Question

Let $X$ be a non-negative random variable. I have an upper bound for $E(1/X)$, for instance, $$ E\left(\frac 1X \right) \le a^{-\alpha n}$$ with $a, \alpha$ positive constants, $n\rightarrow +\infty$. I am interested in finding an upper bound for $$ E\left(\frac 1{X^p} \right)$$ where $p\ge 2$ using the assumption on $E(1/X)$. My question is: there exists an inequality to an upper bound of $E(1/X^p)$ which is related to $E(1/X)$. In this case, the Jensen's just give us an lower bound since $f(E[X]) \le E[f(X)]$. Thank you for any answer.

Answer 1

No. Exercise: For each $p\gt1$, find some sequence $(X_n)$ of positive random variables such that $E\left(\dfrac1{X_n}\right)\to0$ and $E\left(\dfrac1{X_n^p}\right)\geqslant1$ for every $n$. Hint: Each $X_n$ may be two-valued.