On Growth of Polynomials with Restricted Zeros

Abstract

Let $P(z)$ be a polynomial of degree $n$ which does not vanish in $|z|< k $, $k\geq 1$. It is known that for each $0\leq s< n$ and $1\leq R\leq k$, $$M\big(P^{(s)},R\big)\leq \Big(\frac{1}{R^{s}+k^{s}}\Big)\Big[\Big\{\frac{d^{(s)}}{dx^{(s)}}(1+x^{n})\Big\}_{x=1}\Big]\Big(\frac{R+k}{1+k}\Big)^{n}M(P,1).$$ In this paper, we obtain certain extensions and refinements of this inequality by involving binomial coefficients and some of the coefficients of the polynomial $P(z)$.