The mixing advantage for bounded random variables

Corresponding to n independent non-negative random variables X1,...,Xn concentrated on a bounded interval set are values M1,...,Mn, where each Mi is the expected value of the maximum of n independent copies of Xi. We obtain a sharp upper bound for the expected value of the maximum of X1,...,Xn in terms of M1,...,Mn. This inequality is sharp. A similar result is demonstrated for minima