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Inequalities for absolute maxima of a polynomial and its derivative

Authors:

Ramesh V Garimella Yung-Way Liu

Institution:

(1) Department of Mathematics, Tennessee Technological University, Box 5054, Cookeville, TN, 38505, U.S.A.

Abstract:

Suppose z ₁, z ₂, ... z _n are complex numbers with absolute values more than 1 and Arg z _j

Arg z _k for j

k where Arg w stands for the argument of the complex number w in 0,2 pgr

). In this note we show that

$\mathop {{\text{min}}}\limits_{\left| z \right| = 1} \frac{{\left| {\sum\limits_{j = 1}^n {\frac{{z_j }}{{z - z_j }}} } \right|}}{{\left| {\sum\limits_{j = 1}^n {\frac{1}{{z - z_j }}} } \right|}} \geqslant \frac{{\sum\limits_{j = 1}^n {\frac{{\left| {z_j } \right|}}{{\left| {z_j } \right| - 1}}} }}{{\sum\limits_{j - 1}^n {\frac{1}{{\left| {z_j } \right| - 1}}} }}.$

We also give necessary and sufficient conditions for equality in the above inequality. As an application, we improve the result of Govil and Labelle on Bernstein's inequality for some special polynomials.

Keywords:

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