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Markov-type inequality and a lower bound for the moduli of critical values of polynomials. (English. Russian original) Zbl 1284.30003

Dokl. Math. 88, No. 1, 449-450 (2013); translation from Dokl. Akad. Nauk 451, No. 5, 495-497 (2013).
From the text: The following result is announced in this paper.
Theorem 1. For any closed bounded set \(E\) and any polynomial \(P\) of degree \(n \geq 2\),
\[ \mathrm{cap} E \sup_E |P'| \leq \left(\frac{2M}{M_c}\right)^{\frac{1-n}{n}}T'_n\left(T^{\prime-1}_n \left(\frac{M}{M_c}\right)\right)M, \]
where \(T_n(z) = 2^{n-1}z^{n}+\ldots\) ist the Chebyshev polynomial of the first kind of degree \(n\), while \(T^{-1}_n\) is a continuous branch of its inverse function that maps the ray \([0, +\infty)\) to the ray \(\left[\cos\frac{\pi}{2n}, +\infty\right)\).

MSC:

30C10 Polynomials and rational functions of one complex variable
30D15 Special classes of entire functions of one complex variable and growth estimates
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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References:

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