As the titles suggests, I am trying to see if we can upper bound
$\mathbb{E}[X \text{ }| \text{ }X > c] - c$
For now, I am assuming bounded mean on both sides: $0 < m \leq \mathbb{E}[X] \leq M < \infty$, and bounded second moment: $\mathbb{E}[X^2] \leq V \leq \infty$. Additionally, X is a random variable with unbounded support. I have tried using simple bounds like Markov's and Chebyshev's but to no avail. If any further assumptions are needed, I'm also all ears. Any ideas? Thanks!
Edit: to be clear, I am hoping for a somewhat general bound, that is, we can assume nice things about the distribution of X (like being integrable, etc.), but I am looking for a bound for any such distribution, perhaps in terms of $c, F(x), f(x), m, M, V$, etc. I am not sure if I am just missing something from fundamental probability theory.