Abstract
We obtain estimates for the integrals of derivatives of rational functions in multiply connected domains in the plane. A sharp order of growth is found for the integral of the modulus of the derivative of a finite Blaschke product in the unit disc. We also extend the results of 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 results obtained.
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