Suppose we have a random variable $X$ such that $E[|X|^p] < \infty$. Then via Jensen's inequality we know that for any $1 \le q \le p$ \begin{align} E[ |X|^q ] \le E[ |X|^p ]^{\frac{q}{p}} \end{align} Which means that all lower moments a finite as well.
My question is suppose we know what $E[ |X|^q ]$ and $E[ |X|^p ]$ are can we have bound on E[ |X|^r ] for $q \le r \le p$ in terms $E[ |X|^q ]$ and $E[ |X|^p ]$?
Of course we can do what we did before and have $E[ |X|^r ] \le E[ |X|^p ]^{\frac{r}{p}}$ but this bound does not use knowledge of $E[ |X|^q ]$. Intuition suggest that using $E[ |X|^q ]$ should improve the bound.
