Schur flows and orthogonal polynomials on the unit circle

The relation between the Toda lattices and similar nonlinear chains and orthogonal polynomials on the real line has been elaborated immensely for the last decades. We examine another system of the differential-difference equations known as the Schur flow within the framework of the theory of orthogonal polynomials on the unit circle. This system can be exhibited in equivalent form as the Lax equation, and the corresponding spectral measure undergoes a simple transformation. The general result is illustrated on the modified Bessel measures on the unit circle and the long time behavior of their Verblunsky coefficients.Comment: 17 pages, section 4 revised essentiall

On critical points of Blaschke products

We obtain an upper bound for the derivative of a Blaschke product, whose zeros lie in a certain Stolz-type region. We show that the derivative belongs to the space of analytic functions in the unit disk, introduced recently in \cite{FG}. As an outcome, we obtain a Blaschke-type condition for critical points of such Blaschke products.Comment: 6 pages in LaTe