Upperbound on $E(AB)$ where $E(A)\le M < \infty$

Question

Can we find upperbound on $E(AB)$, where $A,B$ are positive random variables with finite moments and $E(A)\le M < \infty$ using $E(B)$? $$ E(AB)\le M E(B) $$ or using some other constant, or how to deal with such problem, when you know upperbound on one expected value and try to do some upperbound of multiplication?

Answer 1

Let $A=B$ be any r.v. with finite first moment and with infinite second moment. For example, we can use random variales with Pareto distribution with shape parameter $\alpha=2$.

Then $\mathbb E(AB)=\mathbb E(A^2)=\infty$ but $\mathbb E(B)<\infty$ and the inequality fails.