On a boolean maximization problem

Among all boolean functions f(j) of n variables, n = 2m + 1, the max(j) min(i) (f(j) (.) x(i)) value of correlation between f(j) and the n independent binary variables x(i), 1 less than or equal to i less than or equal to n, is ((2m)(m)) (.) 2(-2m), which is uniquely achieved by f* = maj (x(1),.... x(n)), the majority decision m function on the xi's. This value is asymptotically less, by a factor of root2/pi approximate to 1.2533, than is achieved by the real vector alpha* which achieves max(\alpha\ = 1) min(alpha (.) x(i))(i) = 1/rootn.