^{2}Department of Mathematics, Semnan University, Semnan, Iran.

Abstract

Let $p(z)=z^s h(z)$ where $h(z)$ is a polynomial of degree at most $n-s$ having all its zeros in $|z|\geq k$ or in $|z|\leq k$. In this paper we obtain some new results about the dependence of $|p(Rz)|$ on $|p(rz)| $ for $r^2\leq rR\leq k^2$, $k^2 \leq rR\leq R^2$ and for $R\leq r \leq k$. Our results refine and generalize certain well-known polynomial inequalities.

A. Aziz, Growth of polynomials whose zeros are within or outside a circle, Bull. Aust. Math. Soc. 35 (1987), no. 2, 247--256.

A. Aziz and Q. G. Mohammad, Growth of polynomials with zeros outside a circle, Proc. Amer. Math. Soc. 81 (1981), no. 4, 549--553.

M. Bidkham and K. K. Dewan, Inequalities for a polynomial and its derivative, J. Math. Anal. Appl. 166 (1992), no. 2, 319--324.

N. K. Govil, On the maximum modulus of polynomials, J. Math. Anal. Appl. 112 (1985), no. 1, 253--258.

A. Mir, On extremal properties and location of zeros of polynomials, PhD Thesis, submitted to Jamia Millia Islamia, New Delhi, 2002.

M. S. Pukhta, Extremal problems for polynomials and on location of zeros of polynomials, PhD Thesis, submitted to the Jamia Millia Islamia, New Delhi, 1995.