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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Bernstein-type inequalities for the derivatives of constrained polynomials


Author: Tamás Erdélyi
Journal: Proc. Amer. Math. Soc. 112 (1991), 829-838
MSC: Primary 41A17
MathSciNet review: 1036985
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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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DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1036985-7
PII: S 0002-9939(1991)1036985-7
Keywords: Markov and Bernstein type inequalities, polynomials with restricted zeros
Article copyright: © Copyright 1991 American Mathematical Society