In the theory of random matrices for unitary ensembles associated with Hermitian matrices, m-point correlation functions play an important role. We show that they possess a useful variational principle. Let μ be a measure with support in the real line, and Kn be the n-th reproducing kernel for the associated orthonormal polynomials. We prove that, for m≥1, det [ K n ( μ , x i , x j ) ] 1 ≤ i , j ≤ m = m ! sup P P 2 ( x ¯ ) ∫ P 2 ( t ¯ ) d μ × m ( t ¯ ) where the supremum is taken over all alternating polynomials P of degree at most n−1 in m variables x¯=(x1,x2,…,xm). Moreover, μ×m is the m-fold Cartesian product of μ. As a consequence, the suitably normalized m-point correlation functions are monotone decreasing in the underlying measure μ. We deduce pointwise one-sided universality for arbitrary compactly supported measures, and other limits.