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Siberian Mathematical Journal

, Volume 55, Issue 1, pp 63–71 | Cite as

On one extremal problem for complex polynomials with constraints on critical values

  • V. N. Dubinin
Article

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.

Keywords

Chebyshev polynomial critical values distortion theorems Markov-type inequalities 

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Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  1. 1.Far-Eastern Federal UniversityVladivostokRussia
  2. 2.Institute of Applied MathematicsVladivostokRussia

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