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

Voir la notice de l'article 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. 
@article{ZNSL_2010_383_a4,
     author = {V. N. Dubinin},
     title = {On the components of the lemniscate containing no critical points of a~polynomial other than its zeros},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {77--85},
     publisher = {mathdoc},
     volume = {383},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2010_383_a4/}
}
TY  - JOUR
AU  - V. N. Dubinin
TI  - On the components of the lemniscate containing no critical points of a~polynomial other than its zeros
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2010
SP  - 77
EP  - 85
VL  - 383
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2010_383_a4/
LA  - ru
ID  - ZNSL_2010_383_a4
ER  - 
%0 Journal Article
%A V. N. Dubinin
%T On the components of the lemniscate containing no critical points of a~polynomial other than its zeros
%J Zapiski Nauchnykh Seminarov POMI
%D 2010
%P 77-85
%V 383
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ZNSL_2010_383_a4/
%G ru
%F ZNSL_2010_383_a4
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/