# generalized Cauchy -Schwarz inequality with characteristic function

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.

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Are X and Y independent? – Did Jul 20 '12 at 14:31
Could you check the equation you are trying to prove? You have both $E$ and $\mathbb{E}$, and there is an $\mathbb{E}(t_1)$ which doesn't make sense. You can use the "edit" button to make corrections. – Nate Eldredge Jul 20 '12 at 14:34
@NateEldredge I am afraid that was the damage I inflicted while doing the edit. My apologies. I have edited it now to correspond to the original. – Sasha Jul 20 '12 at 18:14
X and Y are not independent. The relationship between them is not provided as a condition. – jingsi Jul 21 '12 at 7:45