# Entropy Inequality

I have very hard time to prove the following inequality or to show a contradiction.

$H(X_1,X_2,X_3) + H(X_1,X_2,X_4)+ H(X_1,X_3,X_4) + H(X_2,X_3,X_4) \leq 3(H(X_1,X_2) + H(X_3,X_4))$

The problem is I don't know how to approach the solution.

I would appreciate for any help.

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Are $X_1, X_2...$ random variables? – Henrique Oct 10 '13 at 15:27

Note that $H(X,Y) \leq H(X) + H(Y)$.
$$H(X_1, X_2, X_3) \leq H(X_1, X_2) + H(X_3)\\ H(X_1, X_2, X_4) \leq H(X_1, X_2) + H(X_4)\\ H(X_1, X_3, X_4) \leq H(X_3, X_4) + H(X_1)\\ H(X_2, X_3, X_4) \leq H(X_3, X_4) + H(X_2)$$ Therefore, $$LHS \leq 2(H(X_1, H_2) + H(X_3, H_4)) + H(X_1) + H(X_2) + H(X_3) + H(X_4)$$