Geodesic
Lengths of lemniscates. Variations of rational functions
Sbornik. Mathematics, Tome 198 (2007) no. 8, pp. 1111-1117

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The problem under consideration is the estimate of the length of the lemniscate $$ L(P,r)=\{z:|P(z)|=r^n\}, $$ where $$ P(z)=\prod_{k=1}^{n}(z-z_k),\qquad z_k\in\mathbb C,\quad r>0. $$ It is shown that $|L(P,r)|\le 2\pi n r$. A sharp estimate for the variation of a rational function along a curve of bounded rotation of the secant is also obtained. Bibliography: 15 titles. 
@article{SM_2007_198_8_a2,
     author = {V. I. Danchenko},
     title = {Lengths of lemniscates. {Variations} of rational functions},
     journal = {Sbornik. Mathematics},
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     publisher = {mathdoc},
     volume = {198},
     number = {8},
     year = {2007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2007_198_8_a2/}
}
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V. I. Danchenko. Lengths of lemniscates. Variations of rational functions. Sbornik. Mathematics, Tome 198 (2007) no. 8, pp. 1111-1117. http://geodesic.mathdoc.fr/item/SM_2007_198_8_a2/