The main object of the talk is an entropy function of a probability measure on the unit circle and its relation to orthogonal polynomials and Schur functions. In the first part of the talk we discuss a formula that allows us to evaluate the entropy function knowing the values of Schur functions of a mesure at a given point $z$ of the unit disk. For $z=0$, it coincides with the well-known Szego formula relating the logarithmic integral of a measure and its recurrence coefficients. Then, the entropy function will be used to give a relatively simple proof of the classical theorem by A. Mate, P. Nevai, and V. Totik on averaged convergence of orthogonal polynomials on the unit circle.