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In this paper, we establish some inequalities for a more general class of polynomials P(S(z)), where P(z) = ∑ ajzj and S(z) = ∑ bjzj of degree n and m respectively. These results not only generalize the results due to Dubinin [ J. Math. Sci., 143 (2007), 3069-3076] but also refine a result due to Shah and Liman [ NFAA, 9(2004), 223-232].