Let $X$ be a real sub-exponential r.v. then we can easily show that, for any $t>0$, there exists $K>0$ such that $E[X | X \geq t] \leq t + K $.
My question: is there an equivalence between both properties? Can it be generalized to other classes (like $\Psi_\alpha$ r.v.)? Any pointer to published results would be appreciated.