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

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.