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On Growth of Polynomials with Restricted Zeros

Let P(z) be a polynomial of degree n which does not vanish in |z|<k, k≥1. It is known that for each 0≤ s<n and 1≤R≤k, M ( P,R ) ≤ ( 1 Rs+ks )[{ d(s) dx(s) (1+x) } x=1 ](R+k 1+k 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).