Inequalities for the derivative of a polynomial

Aziz, Abdul

doi:10.1090/S0002-9939-1983-0712634-5

Inequalities for the derivative of a polynomial
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Proc. Amer. Math. Soc. 89 (1983), 259-266 Request permission

Abstract:

Let $P(z) = \sum \nolimits _{j = 0}^n {{a_j}{z^j} = c\prod \nolimits _{j = 1}^n {(z - {z_j})} }$ be a polynomial of degree $n$ and $P’(z)$ its derivative. In this paper we consider the problem of estimating the maximum of $\left | {P’(z)} \right |$ on $\left | z \right | = 1$ under some assumptions on the zeros or on the coefficients of $P(z)$ and obtain certain sharp results.

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