Asymptotic upper bounds for the entropy of orthogonal polynomials in the Szegő class

We give an asymptotic upper bound as n→∞ for the entropy integral, E n (w)=−∫p n 2 (x) log (p n 2 (x))w(x)dx, where p n is the n th degree orthonormal polynomial with respect to a weight w(x) on [−1,1] which belongs to the Szegő class. We also study two functionals closely related to the entropy integral. First, their asymptotic behavior is completely described for weights w in the Bernstein class. Then, as for the entropy, we obtain asymptotic upper bounds for these two functionals when w(x) belongs to the Szegő class. In each case, we give conditions for these upper bounds to be attained.