Geodesic
Several integral estimates of the~derivatives of rational functions on sets of finite density
Sbornik. Mathematics, Tome 187 (1996) no. 10, pp. 1443-1463

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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. 
@article{SM_1996_187_10_a1,
     author = {V. I. Danchenko},
     title = {Several integral estimates of the~derivatives of rational functions on sets of finite density},
     journal = {Sbornik. Mathematics},
     pages = {1443--1463},
     publisher = {mathdoc},
     volume = {187},
     number = {10},
     year = {1996},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1996_187_10_a1/}
}
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V. I. Danchenko. Several integral estimates of the~derivatives of rational functions on sets of finite density. Sbornik. Mathematics, Tome 187 (1996) no. 10, pp. 1443-1463. http://geodesic.mathdoc.fr/item/SM_1996_187_10_a1/