Lower Bound on $E[X Y]$

Question

(Cross-post from math.stackexchange.com Q#166689)

I would like to lower-bound $E[X Y]$ where $X, Y$ are two random variables such that:

1. $X \in [x_0, 1], Y \in [y_0, 1]$

2. $E[X] = x, E[Y] = y$

3. $X \geq Y^k$

Here $x_0, y_0, x > x_0, y > y_0, k$ are known constants. There is the trivial bound $E[X Y] \geq x y_0, \geq x_0 y$. Are any better bounds available?

Answer 1

I'll assume $x_0, y_0 \ge 0$. Presumably $k \ge 0$, since if $k < 0$ the only way to have $X \ge Y^k$ with $0 \le X,Y \le 1$ is $X=Y=1$. Since $X \ge Y^k$, Jensen's inequality says $E[XY] \ge E[Y^{k+1}] \ge y^{k+1}$