Geodesic
On the components of the lemniscate containing no critical points of a polynomial other than its zeros
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 25, Tome 383 (2010), pp. 77-85 
V. N. Dubinin. On the components of the lemniscate containing no critical points of a polynomial other than its zeros. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 25, Tome 383 (2010), pp. 77-85. http://geodesic.mathdoc.fr/item/ZNSL_2010_383_a4/
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     title = {On the components of the lemniscate containing no critical points of a~polynomial other than its zeros},
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Let $P$ be a complex polynomial of degree $n$ and let $E$ be a connected component of the set $\{z\colon|P(z)|\leq1\}$ containing no critical points of $P$ other than its zeros. We prove the inequality $|(z-a)P'(z)/P(z)|\leq n$ for all $z\in E\setminus\{a\}$, where $a$ is the zero of the polynomial $P$ lying in $E$. Equality is attained for $P(z)=cz^n$ and any $z$, $c\neq0$. Bibl. 4 titles.

[1] T. Sheil-Small, Complex polynomials, Cambridge Univ. Press, Cambridge, 2002 | MR | Zbl

[2] V. N. Dubinin,, “O pokrytii vertikalnykh otrezkov pri konformnom otobrazhenii”, Mat. zametki, 28:1 (1980), 25–32 | MR | Zbl

[3] V. N. Dubinin, “Simmetrizatsiya v geometricheskoi teorii funktsii kompleksnogo peremennogo”, Uspekhi mat. nauk, 49:1 (1994), 3–76 | MR | Zbl

[4] W. K. Hayman, Multivalent functions, Cambridge Univ. Press, Cambridge, 1994 | MR | Zbl