Several integral estimates of the derivatives of rational functions on sets of finite density

Majorizing sums of special form are constructed for rational functions and their derivatives $R^{(\mu )}(z)$ (here $\mu =0,1,\dots $, $z \in \mathbb C$). As a consequence, several estimates of $R^{(\mu )}$ in integral metrics are obtained on rectifiable curves $\Gamma$ of finite density $\omega (\Gamma )=\sup \bigl \{\operatorname {mes}_1(\Gamma \cap \Delta )/\operatorname {diam}\Delta \bigr \}$, where the supremum is taken over all open discs $\Delta$. Certain estimates on sets that are not necessarily connected are also obtained.