Kragujevac Journal of Mathematics
University of Kragujevac - Faculty of Science

On Zero Free Regions for Derivatives of a Polynomial

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Authors: M. IBRAHIM, M. ISHFAQ AND N. I. AHMAD WANI

DOI: 10.46793/KgJMat2303.403M

Abstract:

Let P_n denote the set of polynomials of the form

with |a|≤ 1 and |z_k|≥ 1 for 1 ≤ k ≤ n − m. For the polynomials of the form p(z) = z ∏ _k=1ⁿ⁻¹(z −z_k), with |z_k|≥ 1, where 1 ≤ k ≤ n− 1, Brown [?] stated the problem “Find the best constant C_n such that p′(z) does not vanish in |z| < C_n”. He also conjectured in the same paper that C_n = . This problem was solved by Aziz and Zarger [?]. In this paper, we obtain the results which generalizes the results of Aziz and Zarger.

Keywords:

Polynomials, zeros, critical points, derivative, region.

References: