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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^(s),R)≤ (1R^s+k^s )[{d^(s) dx^(s) (1+x^n)}_x=1](R+k1+k)^nM(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). , number=2, journal=Analysis in Theory and Applications, year=2016, month=Apr., pages=181–188