Improved Version of an Inequality for the Derivative of a Polynomial

Abstract

Let p(z) be a polynomial of degree n having no zero in |z| < k, k \(\le\) 1 then Govil [Proc. Nat. Acad. Sci., 50(1980), 50-52] proved

                                                                                            

provided \(\left|p^{\prime}(z)\right|\) and \(\left|q^{\prime}(z)\right|\) attain their maxima at the same point on the circle \(|z|=1\),
where

                                                                          \(q(z)=z^n \overline{p\left(\frac{1}{\bar{z}}\right)}\).

In this paper, we prove a result which improves the above inequality.