IJPAM: Volume 41, No. 5 (2007)

GROWTH OF POLYNOMIALS NOT VANISHING
IN A DISK OF PRESCRIBED RADIUS

A. Aziz

, Q. Aliya

$^{1,2}$ Post Graduate Department of Mathematics
University of Kashmir
Hazratbal Srinagar, Kashmir, 190006, INDIA

e-mail: aaulauzeem@rediffmail.com

Abstract.In this paper, we consider the class of polynomials $P(z)=a_{0}+a_{\mu}z^{\mu}+\cdots+a_{n}z^{n}$ , $1\le\mu\le n,$ of degree n not vanishing in the disk $\vert z\vert<k.$ For $k\ge 1,$ we investigate the dependance of $\text{\rm Max}_{\vert z\vert=1}\vert P(Rz)-P(rz)\vert,~R\ge r\ge 1,$ on $\text{\rm Max}_{\vert z\vert=1}\vert P(z)\vert$ and $\text{\rm Min}_{\vert z\vert=k}\vert P(z)\vert$ . For any given complex number $\beta$ and we also measure the growth of $\text{\rm Max}_{\vert z\vert=1}\vert P(Rz)-\beta P(rz)\vert$ and the growth of $\text{\rm Max}_{\vert z\vert=R}\vert P(\rho z)- P(z)\vert,~\rho >1$ , where $0\le r\le R \le k.$ Our results contitute multifaceted generalizations which besides yielding several interesting results as corollaries also lead to some striking conclusions giving extensions and refinements of some known polynomial inequalities.