Suppose we have a random variable $X$. Is there any way of deriving an upper bound of the following expectation: $$E[X * \mathbf{1}_{X\ge x_0}],$$
where $\mathbf{1}_{()}$ is an 0-1 indicator function, and $x_0$ is a constant.
Specifically, I am trying to see if we can derive a Jensen's inequality-style upper bound, however the original Jensen's inequality certainly doesn't work.
Clearly, $X$ and $\mathbf{1}_{X\ge x_0}$ are positively correlated, and we have $E[X * \mathbf{1}_{X\ge x_0}] \ge E[X]* E[\mathbf{1}_{X\ge x_0}]$, which is the opposite direction that I want to pursue.