On one extremal problem for complex polynomials with constraints on critical values (Q467659)
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scientific article; zbMATH DE number 6365602
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On one extremal problem for complex polynomials with constraints on critical values |
scientific article; zbMATH DE number 6365602 |
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On one extremal problem for complex polynomials with constraints on critical values (English)
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4 November 2014
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Let \(P_n\) be the set of all complex polynomials \(P\) of degree \(n\geq 2\) satisfying \(|P(z)|\leq 1\) whenever \(P'(z)=0.\) Using a symmetrisation method, the author proves a result which gives an upper bound for the product \(|P'(0)P'(1)|\), \(P\in P_n\), in terms of certain functions of the values \(|P(0)|\) and \(|P(1)|\). As an application of this result, a number of sharp inequalities for polynomials is presented. In particular, a new version of a Markov-type inequality for an arbitrary compact set is established.
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Chebyshev polynomial
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critical value
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distortion theorem
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Markov-type inequalities
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symmetrization
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