On $L^1$-estimates of derivatives of univalent rational functions

Abstract

We study the growth of the quantity ${\int }_{\mathbb{T}}|R^{\prime }(z)|dm(z)$ for rational functions $R$ of degree $n$, which are bounded and univalent in the unit disk, and prove that this quantity may grow as $n^{\gamma }$, $\gamma >0$, when $n\to \infty$. Some applications of this result to problems of regularity of boundaries of Nevanlinna domains are considered. We also discuss a related result by Dolzhenko which applies to general (non-univalent) rational functions.

arXiv comment16 pages, to appear in Journal d'Analyse Mathematique