Suppose that $A$ and $B$ are random variables that are correlated, and can only have values in some range $[0,R]$. I would like to lower bound $\mathbb E[A/B]$ the expected value of $A/B$.
I have a concentration bound that says $B < \mathbb E[B] + \epsilon$ with probability $1-\delta$. I would now like to conclude something like
$$\mathbb E\left[\frac{A}{B}\right] \geq \frac{\mathbb E[A]}{\mathbb E[B]+\epsilon}$$
as long as $\delta$ is small enough. Does this line of argument make sense? Does the correlation between $A$ and $B$ make this reasoning not logical?
Also, does there exist a $\delta > 0$ small enough such that this implication holds? I suspect we can concoct an adversarial case where with probability $\delta$, $B = R$. But I'm not sure how it affects things because of the fraction. Any hints?