On the lemniscate components containing no critical points of a polynomial except for its zeros |
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Authors: | V N Dubinin |
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Institution: | 1.Institute of Applied Mathematics,Far-Eastern Branch of the Russian Academy of Sciences,Vladivostock,Russia |
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Abstract: | Let P be a complex polynomial of degree n and let E be a connected component of the set {z : |P(z)| ≤ 1} containing no critical points of P different from its zeros. We prove the inequality |(z − a)P′(z)/P(z)| ≤ n for all z ∈ E \ {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 ≠ 0. Bibliography: 4 titles. |
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