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 will discuss a formula that allows 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. The talk is partially based on joint works with Sergey Denisov.