Abstract
For all fixed complex numbers a and b and a natural n ≥ 2, we study the problem of finding the supremum of the product |P′(0)P′(1)| over the set of all polynomials P of degree n satisfying the following conditions: P(0) = a and P(1) = b, while |P(z)| ≤ 1 for all z for which P′(z) = 0. As an application of the main result of the article, we give a number of exact estimates for polynomials with account taken of their critical values. We in particular establish a new version of a Markov-type inequality for an arbitrary compact set.
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Original Russian Text Copyright © 2014 Dubinin V.N.
The author was partially supported by the Russian Foundation for Basic Research (Grant 13-01-12404-ofi_m2), the Far East Branch of the Russian Academy of Sciences (Grant 12-I-OMN-02), and the Ministry of Education and Science of the Russian Federation (Contract 14.A18.21.0353).
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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 55, No. 1, pp. 79–89, January–February, 2014.
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Dubinin, V.N. On one extremal problem for complex polynomials with constraints on critical values. Sib Math J 55, 63–71 (2014). https://doi.org/10.1134/S003744661401008X
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DOI: https://doi.org/10.1134/S003744661401008X