Bernstein-type inequalities for the derivatives of constrained polynomials

Erdélyi, Tamás

doi:10.1090/S0002-9939-1991-1036985-7

Bernstein-type inequalities for the derivatives of constrained polynomials
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by Tamás Erdélyi

Proc. Amer. Math. Soc. 112 (1991), 829-838

DOI: https://doi.org/10.1090/S0002-9939-1991-1036985-7

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Abstract:

Generalizing a number of earlier results, P. Borwein established a sharp Markov-type inequality on $[ - 1,1]$ for the derivatives of polynomials $p \in {\pi _n}$ having at most $k(0 \leq k \leq n)$ zeros in the complex unit disk. Using Lorentz representation and a Markov-type inequality for the derivative of Müntz polynomials due to D. Newman, we give a surprisingly short proof of Borwein’s Theorem. The new result of this paper is to obtain a sharp Bernstein-type analogue of Borwein’s Theorem. By the same method we prove a sharp Bernstein-type inequality for another wide family of classes of constrained polynomials.

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