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In this paper, we establish some integral mean estimates for polynomials P (z) = a n z n + n υ=μ a n−υ z n−υ , 1 ≤ μ ≤ n, having all its zeros in |z| ≤ k, k ≤ 1. Our results not only generalize and refine some known polynomial inequalities, but also a variety of interesting results can be deduced from these by a fairly uniform procedure.

In this paper we consider for a fixed µ, the class of polynomials P(z) = a0 + Pn ν=µ aνzν, 1 ≤ µ ≤ n, of degree at most n not vanishing in the disk |z| < k, k > 0. For any ρ > σ ≥ 1 and 0 < r ≤ R ≤ k, we investigate the dependence of k P(ρz) − P(σz) kR on k P kr and derive various refinements and generalizations of some well known results.