Abstract
Some methods of numerical analysis, used for obtaining estimations of zeros of polynomials, are studied again, more especially in the case where the zeros of these polynomials are all strictly positive, distinct and real. They give, in particular, formal lower and upper bounds for the smallest zero. Thanks to them, we produce new formal lower and upper bounds of the constant in Markov-Bernstein inequalities in L 2 for the norm corresponding to the Laguerre and Gegenbauer inner products. In fact, since this constant is the inverse of the square root of the smallest zero of a polynomial, we give formal lower and upper bounds of this zero. Moreover, a new sufficient condition is given in order that a polynomial has some complex zeros.
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Draux, A. Improvement of the formal and numerical estimation of the constant in some Markov-Bernstein inequalities. Numerical Algorithms 24, 31–58 (2000). https://doi.org/10.1023/A:1019132924372
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DOI: https://doi.org/10.1023/A:1019132924372