Is there a "reverse" Jensen's inequality up to a constant?

Question

Specifically I was wondering if for fixed $0<p<1$ there exists a constant $C$ such that $(E|X|)^p\leq C_pE(|X|^p)$ for any random variable $X$, giving an inverse of Jensen's inequality. From what I've managed to find it seems the answer might be no and that $C$ might also depend on the range of $X$. I found this paper giving the answer for $|X|\in[m,M]$ with $0<m<M$, but the issue is $C$ depends on $M$ and $m$ and goes to infinity as $m$ goes to zero. So I'm curious is anyone knows if the $C$ must depend on the range or if this dependence has been removed in more recent work or at least improved to only depend on the upper bound.

Answer 1

In order to get a bound like this you need a lower bound on the values of $X$.

Let $X_n$ be a random variable which takes value $0$ with probability $\frac{n-1}{n}$ and value $1$ with probability $\frac{1}{n}$.

Then $E(X_n)^p = (\frac{1}{n})^p$ and $E(X_n^p) = \frac{1}{n}$, so if a universal constant $C_p$ were to exist like you want it would have to satisfy $n^{1-p} \le C_p$ for all $n$, but of course $n^{1-p}$ tends to infinity as $n$ does for $p<1$.