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I am trying to prove or disprove the following inequality.

Let $A,B,C$ be three random variables, such that $$E[A^2] + E[B^2] = 1,$$ $$E[C^2] + E[B^2] = 1,$$ $$E[B(A+C)]= 0,$$ $$0 \leq A,C \text{ and } |B| \leq \sqrt{AC} \text{ almost surely}.$$

Is the following true? $$2 \leq Var(A^2 + B^2) + Var(C^2+B^2) + 2Var(B(A+C)) + E[A]^2 + 2E[B]^2 + E[C]^2$$

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    Counterexample: $A=C=0$ almost surely, $B=\pm1$ symmetric. – 2017-02-26
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    Great example! I've added an extra condition to the inequality to disallow cases like this – 2017-02-26

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