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On maximum modulus for the derivative of a polynomial. (English) Zbl 1190.30003

Summary: If \(P(z)\) is a polynomial of degree \(n\), having all its zeros in the disk \(|z|\leq k\), \(k\geq 1\), then it was shown by N. K. Govil [Proc. Am. Math. Soc. 41, 543–546 (1973; Zbl 0279.30004)] that \[ \underset{|z|=1} {\max}|P'(z)|\geq \frac{n}{1+k^n}\underset{|z|=1} {\max} |P(z)|. \] In this paper, we obtain a generalization as well as an improvement of the above inequality for polynomials of the type \(P(z) = c_0+\sum_{\nu =\mu}^n c_\nu z^\nu\), \(1\leq \mu \leq n\). Also, we generalize a result due to K. K. Dewan and A. Mir [Southeast Asian Bull. Math. 31, No. 4, 691–695 (2007; Zbl 1150.30001)] in this direction.

MSC:

30A10 Inequalities in the complex plane
30C10 Polynomials and rational functions of one complex variable
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
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