Bounds on Expectation $E[A(B-C)^2]$ [This question has been edited for more given conditions].
Given possibly correlated random variables $A,B,C$, I want to find the best upper bound for $E[A(B-C)^2]$ given the following:


*

*$E[A(B-C)]$

*$Cov(A,B), Cov(A,C)$

*$E[B-C]$

*$E[A^k]$, $E[B^k]$, $E[C^k]$ for $k\geq 1$
If it helps, $A$ here is a binary random variable. I have some trivial bounds using problem specific information, but I want to see if there is a universal tight bound that I am missing here.
 A: You have not specified the covariance of $A$ with the higher-order moments of $B$ and $C$, which is going to limit your ability to get a tight bound.  It seems to me that the only thing you can do here is to use the fact that $A$ is binary to impose the bound $0 \leqslant A \leqslant 1$ and you then have:
$$\begin{equation} \begin{aligned}
0 \leqslant \mathbb{E}(A(B-C)^2) 
&\leqslant \mathbb{E}((B-C)^2) \\[6pt]
&= \mathbb{E}(B^2 - 2BC + C^2) \\[6pt]
&= \mathbb{E}(B^2) - 2 \mathbb{E}(BC) + \mathbb{E}(C^2) \\[6pt]
&= \mathbb{E}(B^2) - 2 \mathbb{Cov}(B,C) - 2 \mathbb{E}(B)\mathbb{E}(C) + \mathbb{E}(C^2). \\[6pt]
&\leqslant \mathbb{E}(B^2) + 2 \mathbb{S}(B) \mathbb{S}(C) - 2 \mathbb{E}(B)\mathbb{E}(C) + \mathbb{E}(C^2), \\[6pt]
\end{aligned} \end{equation}$$
where the standard deviations $\mathbb{S}(B)$ and $\mathbb{S}(C)$ can be obtained from the moments you have given.  (For the last step, we note that you have not given sufficient information to get $\mathbb{Cov}(B,C)$, so here we have to use its largest bound in magnitude, given by the square root of the product of the variances.)
A: There different upper bound as you said. For one of them as $A$ is binary, we can say $S = E(A(B-C)^2) \leq p_1E((B-C)^2)$ (Because of values of $A$ is $0$ and $1$). As $p_1$ which the probability of comming $1$ and less than $1$, we can say $S \leq E((B-C)^2) = E(B^2) + E(C^2) - 2E(BC) \leq E(B^2) + E(C^2)$ if $B$ and $C$ are positive.
You can also use from these inequalities to might find some better upper bounds.
