Lengths of lemniscates. Variations of rational functions

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.
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