Besides some trivial lower bound because of the range constraint, there is not much you can say here. I can let Y be X with probability > k for each X value and anything else otherwise. There is a lot of freedom no matter how close k is to 1.

Besides some trivial lower bound because of the range constraint, there is not much you can say here. I can let $Y$ be $X$ with probability $> k$ for each $X$ value and anything else otherwise. There is a lot of freedom no matter how close $k$ is to $1$.