Geodesic
On the Finite-Increment Theorem for Complex Polynomials
Matematičeskie zametki, Tome 88 (2010) no. 5, pp. 673-682

Voir la notice de l'article provenant de la source Math-Net.Ru

For an arbitrary polynomial $P$ of degree at most $n$ and any points $z_1$ and $z_2$ on the complex plane, we establish estimates of the form $$ |P(z_1)-P(z_2)|\ge d_n|P'(z_1)||z_1-\zeta|, $$ where $\zeta$ is one of the roots of the equation $P(z)=P(z_2)$, and $d_n$ is a positive constant depending only on the number $n$.
Mots-clés : complex polynomial
Keywords: finite-increment theorem, Chebyshev polynomial, Zhukovskii function, Markov's inequality, conformal mapping, covering theorem, Steiner symmetrization, conformal capacity. 
@article{MZM_2010_88_5_a3,
     author = {V. N. Dubinin},
     title = {On the {Finite-Increment} {Theorem} for {Complex} {Polynomials}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {673--682},
     publisher = {mathdoc},
     volume = {88},
     number = {5},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2010_88_5_a3/}
}
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V. N. Dubinin. On the Finite-Increment Theorem for Complex Polynomials. Matematičeskie zametki, Tome 88 (2010) no. 5, pp. 673-682. http://geodesic.mathdoc.fr/item/MZM_2010_88_5_a3/