I'm having trouble in proving this inequality which involves characteristic function.
Let $U(t)=\exp\left(i\langle t,X\rangle\right)-\mathbb{E} \left(\exp(i\langle t,X\rangle)\right)$ and $V(s)=\exp\left(i\langle s,Y\rangle\right)-\mathbb{E} \left(\exp(i\langle s,Y\rangle)\right)$
How can I prove this:
$|\mathbb{E}U(t_1)V(s_1)|^2|\mathbb{E}U(t_2)V(s_2)|^2 \le \mathbb{E}|U(t_1)U(t_2)|^2 \mathbb{E}|V(s_1)V(s_2)|^2$
where $\mathbb{E}$ represents the expectation, $|X|^2=X \hat{X}$ represents the conjugate for complex number.