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
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
Copyright © 2006 K. K. Dewan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.