Consider dependent and positive valued random variables $A,B$ and $X$. I want to prove that \begin{equation} E[X^2 A] E[B] \ge E[X A] E[X B]. \end{equation}
If $A$ and $B$ were scalars, above would hold due to Jensen's Inequality. I wonder whether there would be a proof using a similar logic.
Trying a very dependent set of random variables, let $$ (A,B,X)=(1,0,1)\text{ with probability }\tfrac12 $$ and $$ (A,B,X)=(0,1,4)\text{ with probability }\tfrac12 $$ Then $$ E\left[AX^2\right]=\tfrac12\quad\text{and}\quad E[B]=\tfrac12 $$ and $$ E[AX]=\tfrac12\quad\text{and}\quad E[BX]=2 $$ Therefore, $$ E\left[AX^2\right]E[B]=\tfrac14\lt1=E[AX]\,E[BX] $$