  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.
