I am interested in the quantity $$f(X,t) = \int_t^\infty\negthinspace x\ p(x)\ dx,$$ where $p$ is a probability distribution for a positive variable $X$.

1) **Does this quantity $f(X,t)$ have a name**? As the title question suggests, it is a tail of the integral that is cut out when computing the trimmed mean.

2) **Are there any bounds** on $f(X,t)\ / \ f(X,0)$ in terms of $t$ and properties only of $X$ (e.g. moments or quantiles of $X$)? Note $f(X,0)=E[X]$.

Chebyshev's inequality has the right form for a bound on a different quantity, $$\int_t^\infty\negthinspace p(x)\ dx \le \frac{\text{Var}(X)}{(t - E[X])^2}.$$ I am looking for both upper and lower bounds on $f(X,t)$ and Chebyshev's inequality doesn't seem to provide either.

I am especially interested in the case where $X=|Y_1-Y_2|$ where the $Y$'s are i.i.d. variables. Results for the general case might be better-known, and I would appreciate either.