Apologies for the uninformative title.
A paper I am reading states the following fact. If $X$ and $Y$ are independent random variables, assume all well-behaved, with means $\bar{X}, \bar{Y}$ then
$$ E \left[ \max \lbrace \bar{X} \bar{Y} - XY,0 \rbrace \right] \ge E \left[ \max \lbrace \bar{X} \bar{Y} - \bar{X}Y,0 \rbrace \right].$$
As a reference they give this and call this 'a well known finding in martingale theory'. But that's an entire book and I can't seem to find this result.
Any assistance finding or giving the proof, or explaining the intuition, would be helpful!
The function $f(z)=\max(z,0)$ is convex. Let $Z:=\bar{X} \bar{Y} - XY$. Then by the conditional version of Jensen's inequality (see [1] or [2] or [3]), $$E \left[ \max \lbrace \bar{X} \bar{Y} - XY,0 \rbrace |Y \right] =E[f(Z)|Y] \ge f\bigl(E[Z|Y]\bigr)=\max\{\bar{X} \bar{Y} - \bar{X}Y,0\} \,.$$ Taking expectations of both sides (using the tower property of conditional expectation) proves the requested inequality.
[1] Theorem 3 in https://www.uio.no/studier/emner/matnat/math/MAT4410/h19/undervisningsmateriale/jensen.pdf
[2] David Williams, probability with Martingales, Cambridge University Press.
[3] https://web.stat.tamu.edu/~suhasini/teaching673/conditioning.pdf Page 153