statistics - Inequality for inverse random variable

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.