Asymptotics of Orthogonal Polynomials Beyond the Scope of Szegő's Theorem

First, we give a simple proof of a remarkable result due to Videnskii and Shirokov: let $B$ be a Blaschke product with $n$ zeros; then there exists an outer function $\phi$, $\phi(0)=1$, such that $\|(B\phi)'\|\le C n$, where $C$ is an absolute constant. Then we apply this result to a certain problem of finding the asymptotics of orthogonal polynomials.