Journal of Inequalities and Applications 
Volume 2006 (2006), Article ID 54816, 6 pages
doi:10.1155/JIA/2006/54816

On the derivative and maximum modulus of a polynomial

K. K. Dewan,1 N. K. Govil,2 Abdullah Mir,1 and M. S. Pukhta1

 1Department of Mathematics, Jamia Millia Islamia, New Delhi 110025, India
2Department of Mathematics, Auburn University, Auburn 36849-5310, AL, USA
Received 2 October 2004; Accepted 11 October 2005

Abstract

If p(z)=∑v=0navzv is a polynomial of degree n, having all its zeros in |z|≤1, then it was proved by Turán that |p′(z)|≥(n/2)max|z|=1|p(z)|. This result of Turán was generalized by Govil, who proved that if p(z) has all its zeros in |z|≤K, K≥1, then max|z|=1|p′(z)|≥(n/(1+Kn))max|z|=1|p(z)|, K≥1. In this paper, we sharpen this, and some other related results.