I am trying to show whether the following statement is true or not:
$E(X^2|A)E(1|A) \ge E(X|A)^2$
It is straightforward to prove this statement if $X$ was not conditioned on event $A$. Because, one can simply show
$E(X^2)E(1) = E(X^2)$ and $E(X^2)-E(X)^2 = E(X-E(X))^2 \ge 0 $.
However, I am not sure whether the same logic would apply to the conditional expectations or not.