  We give a simple proof of a classical theorem by A.M\'at\'e, P.Nevai, and
V.Totik on asymptotic behavior of orthogonal polynomials on the unit circle. It
is based on a new real-variable approach involving an entropy estimate for the
orthogonality measure. Our second result is an extension of a theorem by
G.Freud on averaged convergence of Fourier series. We also discuss some related
open problems in the theory of orthogonal polynomials on the unit circle.
