$L^1$-estimates of derivatives of univalent rational functions and related topics

In the talk we consider the growth of the quantity $\int_{\mathbb T}|R'(z)|\,dm(z)$ for the rational functions $R$ of degree $n$, which are bounded and univalent in the unit disk. It will be shown that this quantity may grow as $n^\rho$, $\rho>0$, when $n\to\infty$. The estimates of the number $\rho$ will be presented and some applications will be considered. It is also planned to discuss a related result by Dolzhenko which applies to general (non-univalent) rational functions.