Implications of summing and multiplying inequalities Consider the discrete random variables $X,Y,Z, W$ respectively with supports $\{x_1, x_2\}$, $\{y_1, y_2\}$, $\{z_1, z_2, z_3\}$, $\{w_1, w_2, w_3\}$.
Assume
$$
Pr(X=x_1)\leq Pr(Z=z_1)+Pr(Z=z_2) \hspace{1cm} [a]
$$
$$
Pr(X=x_2)\leq Pr(Z=z_3)+Pr(Z=z_2) \hspace{1cm} [b]
$$
$$
1=Pr(X=x_1)+Pr(X=x_2)= Pr(Z=z_1)+Pr(Z=z_2)+Pr(Z=z_3) \hspace{1cm} [c]
$$
$$
Pr(Y=y_1)\leq Pr(W=w_1)+Pr(W=w_2) \hspace{1cm} [d]
$$
$$
Pr(Y=y_2)\leq Pr(W=w_3)+Pr(W=w_2) \hspace{1cm} [e]
$$
$$
1=Pr(Y=y_1)+Pr(Y=y_2)= Pr(W=w_1)+Pr(W=w_2)+Pr(W=w_3) \hspace{1cm} [f]
$$
I want to show that the above inequalities imply that
$$
(\star) \hspace{1cm}
Pr(X=x_1)*Pr(Y=y_1)+Pr(X=x_2)*Pr(Y=y_2)
$$
$$
\leq \Big[Pr(Z=z_1)+Pr(Z=z_2)\Big]*\Big[Pr(W=w_1)+Pr(W=w_2)\Big]
$$
$$
+\Big[Pr(Z=z_3)+Pr(Z=z_2)\Big]*\Big[Pr(W=w_3)+Pr(W=w_2)\Big]
$$
$$
-Pr(Z=z_2)*Pr(W=w_2)
$$

My thoughts: by mixing [a], [b], [d], [e] it is easy to show that $$
Pr(X=x_1)*Pr(Y=y_1)+Pr(X=x_2)*Pr(Y=y_2)
$$
$$
\leq \Big[Pr(Z=z_1)+Pr(Z=z_2)\Big]*\Big[Pr(W=w_1)+Pr(W=w_2)\Big]
$$
$$
+\Big[Pr(Z=z_3)+Pr(Z=z_2)\Big]*\Big[Pr(W=w_3)+Pr(W=w_2)\Big]
$$
Nevertheless, it seems to me impossible to show that ($\star$) holds. If you could provide a set of numbers satisfying [a], [b], [c], [d], [e], [f] but violating ($\star$), I would gratefully accept that as an answer.

For convenience, $x_1$ is used to denote $\Pr(X = x_1)$ and similarly for other probabilities. We have
$$
(x_1 + x_2)(y_1 + y_2) = (w_1 + w_2 + w_3) (z_1 + z_2 + z_3)
$$
from which we obtain
$$
\small
x_1y_1 + x_2y_2 + \color{red}{x_2y_1 + x_1y_2} = (w_1 + w_2)(z_1 + z_2) + (w_3 + w_2)(z_3 + z_2) - z_2w_2 + \color{red}{z_3w_1 + w_3z_1} \tag{$\spadesuit$}
$$
Since $x_1 \leq z_1 + z_2$, therefore
$$
x_2 = 1 - x_1 \geq 1 - z_1 - z_2 = z_3
$$
Similarly,
$$
x_1 \geq z_1, \quad y_2 \geq w_3, \quad\text{and}\quad y_1 \geq w_1
$$
Thus
$$
x_2y_1 + x_1y_2 \geq z_3w_1 + z_1w_3
$$
where the LHS and the RHS are those red in $(\spadesuit)$. We conclude that
$$
x_1y_1 + x_2y_2 \leq (w_1 + w_2)(z_1 + z_2) + (w_3 + w_2)(z_3 + z_2) - z_2w_2
$$
