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We give an asymptotic upper bound as $n\to\infty$ for the entropy integral$$E_n(w)= -\int p_n^2(x)\log (p_n^2(x))w(x)dx,$$ where $p_n$ is the $n$thdegree orthonormal polynomial with respect to a weight $w(x)$ on $[-1,1]$ whichbelongs to the Szeg\H{o} class. We also study two functionals closely relatedto the entropy integral. First, their asymptotic behavior is completelydescribed for weights $w$ in the Bernstein class. Then, as for the entropy, weobtain asymptotic upper bounds for these two functionals when $w(x)$ belongs tothe Szeg\H{o} class. In each case, we give conditions for these upper bounds tobe attained.Comment: 17 page

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