  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 $n$th
degree orthonormal polynomial with respect to a weight $w(x)$ on $[-1,1]$ which
belongs to the Szeg\H{o} 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 Szeg\H{o} class. In each case, we give conditions for these upper bounds to
be attained.
