Estimates for integrals of derivatives of rational functions in multiply connected domains in the plane

TL;DR

This paper derives sharp estimates for integrals of derivatives of rational functions in multiply connected planar domains, extending previous results to broader classes of domains with rectifiable boundaries.

Contribution

It generalizes existing estimates for derivatives of rational functions to multiply connected domains with rectifiable boundaries, including sharpness analysis.

Findings

Sharp growth order for derivatives of finite Blaschke products in the unit disk.

Extension of Dolzhenko's results to wider classes of domains.

Validation of the sharpness of the obtained estimates.

Abstract

We obtain estimates for integrals of derivatives of rational functions in multiply connected domains in the plane. A sharp order of the growth is found for the integral of the modulus of the derivative of a finite Blaschke product in the unit disk. We also extend the results of E.P. Dolzhenko about the integrals of the derivatives of rational functions to a wider class of domains, namely, to domains bounded by rectifiable curves without zero interior angles, and show the sharpness of the obtained results.