Let us consider a real-valued r.v. $X$ such that $\mathsf E X > y$ and $X\geq x_0$ a.s. Denote $ g(x,y,r) = \frac{1}{y+r} - \frac{1}{x+r}. $
It is right that for any $X$ there exists $r_0>-x_0$ such that for all $r>r_0$ one have $\mathsf E g(X,y,r)>0$?
The question is similar to this one: Estimation of an expectation
I guess it can be done through the expansion in series. Anyway if there will be another idea - I appreciate it since expansion will give $r_0$ which increases with $y$.