Dictator functions maximize mutual information

Let (..,..) denote n independent, identically distributed copies of two arbitrarily correlated Rademacher random variables (..,..).We prove that the inequality I(f(..);g(..)).I(..;..)holds for any two Boolean functions: f,g:{.1,1}n.{.1,1};denotes mutual information].We further show that equality in general is achieved only by the dictator functions f(x)=±g(x)=±xi, i.{1,2,.,n}.