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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Inequalities for polynomials with a prescribed zero

Authors: A. Giroux and Q. I. Rahman
Journal: Trans. Amer. Math. Soc. 193 (1974), 67-98
MSC: Primary 30A06
MathSciNet review: 0352427
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Abstract: Inequalities for the derivative and for the maximum modulus on a larger circle of a polynomial with a given zero on the unit circle are obtained in terms of its degree and maximum modulus on the unit circle; examples are given to show that these are sharp with respect to the degree (best constants are not known). Inequalities for ${L^p}$ norms, in particular ${L^2}$ norms, are also derived. Also certain functions of exponential type are considered and similar inequalities are obtained for them. Finally, the problem of estimating ${P_n}(r)$ (with $0 < r < 1$) given ${P_n}(1) = 0$ is taken up.

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Keywords: Derivative of a polynomial, Bernstein’s inequality, growth of maximum modulus, problem of Hal&#225;sz, <IMG WIDTH="29" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${L^p}$"> norm of a polynomial, trigonometric polynomial, entire function of exponential type, interpolation formula
Article copyright: © Copyright 1974 American Mathematical Society